CHAPTER 3. Number Theory
|
|
- Paula Small
- 5 years ago
- Views:
Transcription
1 CHAPTER 3 Number Theory 1. Factors or not According to Carl Friedrich Gauss ( ) mathematics is the queen of sciences and number theory is the queen of mathematics, where queen stands for elevated and beautiful. Number theory is mainly the study of the system of integers Z = {0, ±1, ±2,...} and the consequences of the fact that division is not always possible within Z. E.g. 15/3 Z but 15/4 / Z. Let us start with the well-nown long division that has rather surprising consequences. Example (= ) (= ) (= 6 223) What is really being done and what has been achieved? On the right some zeros are filled in that are not written on the left. We now see that = 45. In effect, multiples of 223 were subtracted from until 223 could not be subtracted anymore without running into negative numbers. The mathematical content of long division is the following theorem and long division is just an efficient way of finding q and r in the Division Theorem. 1
2 2 3. NUMBER THEORY Theorem 1.2. Division Theorem. Given integers a, b, b > 0, there exist unique integers q and r such that a = qb + r, 0 r < b The quotient q and the remainder r can be found by repeated subtraction of b. The division algorithm is just an efficient way for computing q and r by repeated subtractions. Example 1.3. Let a = and b = 735. Find non-negative integers q and r such that = 735q + r and 0 r < 735. Answer: By Long Division we find q = 772 and r = 503. Example 1.4. Let a = and b = Find non-negative integers q and r such that = 7593q + r and 0 r < Answer: By Long Division we find q = 429 and r = 0. Definition 1.5. An integer a is even if a = 2x for some integer x, i.e., r = 0 in Theorem 1.2. An integer b is odd if b = 2x + 1 for some integer x in Z, i.e., r = 1 in Theorem 1.2. Proposition 1.6. Every integer is either even or odd. The product of two odd integers is odd. Proof. Let a and b be odd integers. According to the definition of odd integer there exist integers x and y such that a = 2x + 1 and b = 2y + 1. Then ab = (2x + 1)(2y + 1) = (2x + 1)(2y) + (2x + 1) 1 = 2(y(2x + 1)) + 2x + 1 = 2(y(2x + 1) + x) + 1. Here z = y(2x+1) + x is an integer by the closure properties of Z, hence ab = 2z +1 is an odd number. Theorem is not rational. Proof. (Aristotle) By way of contradiction assume that 2 = a/b where either a or b is odd. Then 2b 2 = a 2, hence a 2 is even and therefore a is even. This means that a = 2a and substituting 2b 2 = 4a 2. Thus b 2 = 2a 2. This says that b is even, a contradiction. Definition 1.8. (1) Let a, f be integers. We say that f is a factor of a if a = f some integer or a = f a for some integer a. (2) Let a and b be given integers. An integer f is a common factor of a and b if f is a factor of a and a factor of b.
3 1. FACTORS OR NOT 3 (3) The greatest common factor of two integers a and b is the largest among the common factors of a and b. The greatest common factor of a and b is denoted by gcf(a, b). Remar 1.9. In the literature it is much more common to say f divides a (or f divides a evenly) than to say that f is a factor of a. However, the first gets confused with other uses of divide and therefore we will avoid its use. Remar Suppose that a and b are positive integers. Then b is a factor of a if and only if in the Division Theorem a = qb + r the remainder r = 0. Therefore instead of saying b is a factor of a it is also said that b divides a evenly. Factoring Rules. Let a be a positive integer given in base 10 representation. Then the following rules are true. (1) 2 is a factor of a if its units digit is even. (2) 3 is a factor of a if 3 is a factor of the sum of the digits of a. (3) 9 is a factor of a if 9 is a factor of the sum of the digits of a. Recall that gcf(a, b) is the largest among the common factors of a and b. Therefore gcf(a, b) can be found in the following way which is instructive but not very efficient. Remar Finding greatest common factors Let a and b be given integers. (1) List the positive factors of both a and b. (2) List the common (positive) factors of a and b. (3) Pic the largest in the list of common factors. Example Finding greatest common factors. Let a = 12 and b = 28. (1) The positive factors of 12 are 1, 2, 3, 4, 6, 12. (2) The positive factors of 28 are 1, 2, 4, 7, 14, 28. (3) The common factors are 1, 2, 4. (4) The greatest common factor is 4. Example Let a = 240 and b = 330. The positive factors 0f 240 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20,24, 30, 40, 48,60, 80, 120, 240, and the positive factors of 330 are 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55,66,110,165, 330. The positive common factors of a and b are 1, 2, 3, 5, 6, 10, 15, 30, and the greatest common factor of a and b is 30, gcf(240, 330) = 30. The following lemmas will be used over and over again and should be memorized. Lemma Let a, b, c be integers and suppose that c = a + b. If a number f is a factor of two of the three integers a, b, c, then f is a factor of the third.
4 4 3. NUMBER THEORY Lemma Let a and b be integers. If f is a factor of a, then f is a factor of ab. The Euclidean Algorithm is a very beautiful and efficient way of finding the greatest common factor of two integers. It is bases on the following fact. Lemma Let a and b be integers and a = bq +r for some integers q and r. Then the common factors of a and b are exactly the same as those of b and r. In particular, gcf(a, b) = gcf(b, r). Example Let a = and b = Then a = 16 b + 755, hence Further, 3476 = , so gcf(56371, 3476) = gcf(3476, 755). gcf(56371, 3476) = gcf(3476, 755) = gcf(755, 456). Further, 755 = , 456 = , 299 = , 157 = , hence gcf(56371, 3476) = gcf(755, 456) = gcf(456, 299) = gcf(299, 157) = gcf(157, 122) = gcf(122, 35). It is clear that 35 has the positive factors 1, 5, 7, 35 and of these only 1 is a factor of 122. Hence gcf(56371, 3476) = 1. Theorem Let a and b be (positive) integers. Then there exist integers u and v such that gcf(a, b) = ua + vb. Consequently, every common factor of a and b is a factor of gcf(a, b). The integers u, v and gcf(a, b) can be found efficiently using the Euclidean Algorithm described below. Example Let a = and b = 347. Then gcf(69321, 347) = gcf(a, b) = ua + vb where gcf(a, b) = 1, u = 36 and v = Proof. (1) = (2) 347 = (3) = (1) 1640 (2) 241 = (4) = (2) 1 (3) 106 = (5) = (3) 2 (4) 29 = (6) = (4) 3 (5) 19 = (7) = (5) (6) 10 = (8) = (6) (7) 9 = (9) = (7) (8) 1 =
5 The long divisions used: 1. FACTORS OR NOT = = = = = = = The same process can be done in a short form that only lists the essential data. action a b (1) (2) (3) = (1) 1640 (2) (4) = (2) 1 (3) (5) = (3) 2 (4) (6) = (4) 3 (5) (7)=(5)-(6) (8)=(6)-(7) (9)=(7)-(8) Example Let a = 3675 and b = 791. Then gcf(a, b) = 7 = Proof. action a b (1) (2) (3) = (1) 4 (2) (4) = (2) 1 (3) (5) = (3) 1 (4) (6) = (4) 4 (5) (7) = (5) 1 (6) (8) = (6) 1 (7) (9) = (7) 2 (8) Definition An integer m is a multiple of the integer a if a is a factor of m. An integer m is a common multiple of the integers a and b if m is a multiple of both a and b. The smallest positive common multiple of a and b is the least common multiple of a and b denoted by lcm(a, b).
6 6 3. NUMBER THEORY Given two fractions a/b and c/d, the least common denominator of the fractions is lcm(b, d). Example Let a = 35 and b = 21. Some multiples of a (there are infinitely many of them) are a = 35, 70, 105, 140, 175,... and some multiples of b are b = 21, 42, 63, 84, 105, 126,... We can now see that lcm(35, 21) = 105. It is easy to see that gcf(35, 21) = 7, and we note the curious fact that gcf(a, b) lcm(a, b) = = 735 = ab. Theorem For any postive integers a and b, gcf(a, b) lcm(a, b) = ab. Remar To find the least common multiple of two integers a and b, we use the Euclidean algorithm to find gcf(a, b) and then compute lcm(a, b) = a b. gcf(a,b) Example (1) lcm(240, 330) = = = (See Example 1.13). 30 (2) lcm(56371, 3476) = = = (See Example 1.17). 1 (3) lcm(569321, 347) = = (See Example 1.19). (4) lcm(3675, 791) = = = (See Example 1.20). 7 Example = = The Fundamental Theorem of Arithmetic Given a positive integer, say 113, it can always be factored as 113 = 1 113, in general a = 1 a = a 1. This is an uninteresting trivial factorization. Definition 2.1. An (positive) integer a is composite if it can be factored as a = b c where neither b nor c is equal to one. In other words, the (positive) integer a is composite if it is the product of two positive integers that are both smaller than a. A positive integer that is not composite - can only be factored trivially - is a prime number or simply a prime. Lemma 2.2. Let p be a prime. Then p has exactly two positive factors, namely 1 and p. Let a be any other integer. Then gcf(a, p) = 1 or gcf(a, p) = p depending on whether p is a factor of a or not.
7 2. THE FUNDAMENTAL THEOREM OF ARITHMETIC 7 Example 2.3. The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97 are all prime numbers. The number 123 is composite because 123 = The number 3127 is composite because 3127 = The number 1111 is composite because 1111 = Proposition 2.4. Let n be a positive integer and suppose that n = a b where a and b are again positive integers. If b n, then a n. Proof. If b n and also a > n then a b > n n = n while a b = n by hypothesis. Corollary 2.5. If the positive integer p has no factors that are less than or equal to p then p is a prime number. Proof. If p had a factor larger than n then it would have a factor n and these have all been eliminated. Corollary 2.6. If the positive integer p has no prime factors which are less than or equal to p then p is a prime number. Proof. If p has no prime factors n then there is no composite factor n either, since such a factor would in turn contain a prime factor of n. Theorem 2.7. There exist infinitely many primes. Proof. This is an example of a proof by contradiction. We mae an assumption, then derive an impossibility from it and conclude that the assumption was false. Assume that there are only finitely many primes. List them in increasing order: (2.8) 2, 3, 5, 7, 11, 13, 17,..., P, so P is the largest prime. Study the number n obtained by multiplying all the primes and adding 1: n = p P + 1. Now 2 is not a factor of n since 2 is not a factor of 1; 3 is not a factor of n since 3 is not a factor of 1; 5 is not a factor of n since 5 is not a factor of 1; in general, p is not a factor or n because p is not a factor of 1. So n is either a prime itself different from any one in the list (2.8) or it is a product of primes, each of which is not contained in the list (2.8). This means that the list (2.8) did not contain all the primes after all, so the assumption that there are only finitely many primes is false. Therefore, there are infinitely many primes. Theorem 2.9. Fundamental Theorem of Arithmetic. Every integer a > 1 can be factored uniquely as a = p n 1 1 pn 2 2 pn,
8 8 3. NUMBER THEORY n i 1, p 1 < p 2 < < p, p i primes. We will indicate a proof later. For the moment we observe that every integer a > 1 is a product of primes for the following reason. If a is itself a prime it is considered a product of primes. If a is composite, then a = a 1 a 2 and both a 1 and a 2 are smaller than a. Now loo at a 1 and a 2. Factor them if they are composite, and eep going until you arrive at primes. The process have to come to a halt because integers cannot become smaller and smaller forever. Example = = (11 49) (17 17) = 11 (7 7) = = Before getting into the uniqueness proof, let us see what it can do for us. The uniqueness says that however we arrive at a factorization into a product of primes, the result is always the same. Example Let a = Now suppose that b is a factor of a and a = b c. Let b = p n 1 1 p n and c = p m 1 1 p m, where p 1 < p 2 < < p, and 0 n i, m i, be the prime factorizations of b and c as in Theorem 2.9. Now a = b c = p n 1+m 1 1 p n +m. By the uniqueness of the prime factorization we must have a = = p n 1+m 1 1 p n +m. This means that p 1 = 2, n 1 + m 1 = 3, p 2 = 5, n 2 + m 2 = 2, p 3 = 7, and n 3 + m 3 = 1. We conclude that = 3, n 1 3, n 2 2 and n 3 1. Using this we can list the positive factors of a = = 1400 (there are = 24 of them) as follows: = 1, = 7, = 5, = 35, = 25, = 175, = 2, = 14, = 10, = 70, = 50, = 700, = 4, = 28, = 20, = 140, = 100, = 700, = 8, = 56, = 40, = 280, = 200, = Corollary Let a = p n 1 1 pn 2 2 pn, n i 1, p 1 < p 2 < < p, p i primes. Then the positive factors of a are exactly the integers f = p i 1 1 p i 2 2 p i, where 0 i j n i. There are (n 1 + 1)(n 2 + 1) (n + 1) such factors.
9 2. THE FUNDAMENTAL THEOREM OF ARITHMETIC 9 Example How many positive factors does 120 posses? We factor 120 = There are = 16 factors. Example (1) The number a = has = 315 different positive factors. (2) The number 512 = 2 9 has 10 different positive factors. (3) The number = has 64 different positive factors. We come to a crucial result. Theorem Let a, b, c be integers. If a is a factor of bc and gcf(a, b) = 1, then a is a factor of c. Proof. It is given that gcf(a, b) = 1. By Theorem 1.18 there are integers u, v such that 1 = ua + vb. Multiplying the equation by c we obtain c = uac + vab. Now a is a factor of uac trivially, and c is a factor of ab, so vab by hypothesis. By Lemma 1.14 a is a factor of c. Corollary Let p, p 1, p 2 primes. If p is a factor of p 1 p 2, then p = p 1 or p = p 2. Corollary Let p, p 1,...,p n be primes. If p is a factor of p 1 p 2 p n, then p = p 1 or p = p 2, or... or p = p n. Corollary 2.17 is the result that is needed to prove the uniqueness part of the Fundamental Theorem of Arithmetic. We illustrate the formal proof by an example. Example We find that = Somebody else come up with a mysterious factorization (2.19) = p n 1 1 p n 2 2 p n, p 1 < p 2 < < p. Therefore we have that = p n 1 1 p n 2 2 p n Tae a factor p i of the right hand side which is then also a factor of the left hand side. By Corollary 2.17 p i = 2 or p i = 3 or p i = 5 or p i = 7. Hence (2.19) must loo lie = 2 n 1 3 n 2 5 n 3 7 n 4 and we have = 2 n 1 3 n 2 5 n 3 7 n 4
10 10 3. NUMBER THEORY Now 2 is a factor of the left hand side so it is a factor of the right hand side by Corollary This means that n 1 1 and we can cancel 2 to get = 2 n n 2 5 n 3 7 n 4 The prime 2 is still a factor of the left hand side, so of the right hand side, and we must have n This means we can cancel another 2 and obtain = 2 n n 2 5 n 3 7 n 4. Now there is no 2 on the left so there cannot be a 2 on the right and we conclude that n 1 2 = 0 or n 1 = 2. We next loo at the prime 3 that appears on the left. It must appear on the right also, so n 2 1. Canceling the 3 we have = 3 n n 3 7 n 4. We conclude that n 2 = 1 because no 3 appears on the left and have now = 5 n 3 7 n 4. We have a five on the left, so n 3 1 enabling us to cancel a 5 and get = 5 n n 4. We still have a five on the left, so n enabling us to cancel another 5 to get 5 7 = 5 n n 4. We still have a five on the left, so n enabling us to cancel another 5 to get 7 = 5 n n 4. We finally get n 3 = 3 and n 4 = 1 showing that the mysterious factorization is identical with ours. Proposition Common Factors. Suppose a = p n 1 1 pn 2 2 pn pn is the factorization of the integer a into a product of primes, and b = p m 1 1 p m 2 2 p m p m is the factorization of the integer b into a product of primes, then the common factors of a and b are all integers f = p E 1 1 pe 2 2 pe p E where E i is less than or equal to both n i and m i. Therefore the greatest common factor of a and b is gcf(a, b) = p e 1 1 p e 2 2 p e p e where e i is the lesser of n i and m i. Proposition Multiples. (1) The positive multiples of an integer a are the integers a b where b is any positive integer.
11 2. THE FUNDAMENTAL THEOREM OF ARITHMETIC 11 (2) If a = p n 1 1 pn 2 2 pn pn is the factorization of a into a product of primes, then maing the exponents n i larger produces multiples, also bringing in additional primes produces multiples. (3) The multiples of a are all integers of the form where m i n i and s i 0. m = p m 1 1 pm 2 2 pm p m qs 1 1 qs 2 2 qs 3 3 Proposition Common Multiples. Suppose a = p n 1 1 p n 2 2 p n p n is the factorization of the integer a into a product of primes, and b = p m 1 1 p m 2 2 p m p m is the factorization of the integer b into a product of primes, then the common multiples of a and b are all integers m = p M 1 1 p M 2 2 p M p M qs 1 1 qs 2 2 qs 3 3 where M i is greater or equal to both n i and m i and s i 0. Therefore the least common multiple of a and b is where M i is the greater of n i and m i. lcm(a, b) = p M 1 1 p M 2 2 p M p M Theorem Let a and b be positive integers. Then Example Let and let Then and and hence lcm(a, b) gcf(a, b) = a b. a = b = lcm(a, b) = gcf(a, b) = lcm(a, b) gcf(a, b) = ( )( ) = ab.
12 12 3. NUMBER THEORY 3. Exercises Exercise 3.1. For each of the following values of a and b find the unique integers q and r such that a = qb + r, 0 r < b. (1) a = 723, b = 23. (2) a = 1582, b = 231. (3) a = , b = (4) a = 12345, b = (5) a = 0, b = 13. (6) a = 365, 904, b = Exercise 3.2. Decide whether or not the following numbers have factors of 2, 3, 4, 5 and , , , 29480, , Exercise 3.3. For the following integers a and b list the positive factors and find the greatest common factor gcf(a, b). (1) a = 115, b = 225. (2) a = 1111, b = 333. (3) a = 237, b = (4) a = 17 23, b = Exercise 3.4. Find the greatest common factor of the following integers a and b using the method of Euclid. (1) a = 543, b = 113. (2) a = 4563, b = 981. (3) a = 451, b = 85. (4) a = 1111, b = (5) a = 12345, b = Exercise 3.5. Find the greatest common factor of the following integers a and b and find integers u, v such that gcf(a, b) = ua + vb. (1) a = 543, b = 113. (2) a = 4563, b = 981. (3) a = 451, b = 85. (4) a = 1111, b = (5) a = 12345, b = Exercise 3.6. Find the least common multiple of the following integers a and b by listing common multiples and looing for the least one among these. (1) a = 543, b = 113. (2) a = 56, b = 98. (3) a = 45, b = 85.
13 (4) a = 111, b = 111. (5) a = 123, b = EXERCISES 13 Exercise 3.7. Find the least common multiple of the following integers a and b using Theorem (1) a = 543, b = 112. (2) a = 4563, b = 981. (3) a = 451, b = 85. (4) a = 1111, b = (5) a = 12345, b = Exercise 3.8. Compute the following sums and differences using least common denominators. 28 (1) (2) (3) (4) (5) Exercise 3.9. Which ones of the following numbers are prime and which ones are composite? 211, 373, 453, 565, 463, 371, 637, 343, , Exercise How many positive factors do the following integers possess? 24, 67, 69, 2445, 1111, 999, 5 6, p 7 where p is a prime, 7 n, p n where p is a prime. Exercise Using your calculator find integers q and r such that = q r, 0 r < Exercise Let x and y be (unnown) integers that are related by the equality x 130 = 75y. Why are the following statements true? (1) 5 is a factor of x. (2) 3 is not a factor of x. (3) If y has a factor of 13, then x has a factor of 13.
14 14 3. NUMBER THEORY (4) If y is even, then x is also even. (5) If y is odd, then x is also odd. (6) If gcf(13, y) = 1, then 13 cannot be a factor of x. Exercise Verify the following statements. (1) odd odd = odd. (2) even even = even. (3) even odd = even. (4) odd + odd = even. (5) even + even = even. Exercise Let a = b where b is some unnown positive integer such that gcf(2 5 19, b) = 1. Which of the following numbers are factors of a and which ones are not? 16, 80, 95, 361, 100, 76. Exercise The integer 378 has 16 different positive factors and a partial list of factors is 1, 2, 3, 6, 7, 9, 14, 18, 21, 63, 378. Complete the list. 4. Solution to exercises 3.1 Double chec your answers by computing qb + r. It has to come to be a has factors 3 and 5, has factor 2, has factor 5, has factors 4 and 5, has a factor 2, has a factor of gcf(115, 225) = 5, 115 = 5 23, 225 = gcf(1111, 333) = 1, 1111 = , 333 = gcf(237, 1659) = 237, 237 = 3 79, 1659 = gcf(17 23, 17 19) = gcf(543, 113) = 1; gcf(4563, 981) = 9; gcf(451, 85) = 1; gcf(1111, 11111) = 1; gcf(12345, 1234) = In the following the triples (,, ) are the rows of the tables in the text. (543, 1, 0) 4(113, 0, 1) = (91, 1, 4) (113, 0, 1) (91, 1, 4) = (22, 1, 5) (91, 1, 4) 4 (22, 1, 5) = (3, 5, 24) (22, 1, 5) 7 (3, 5, 24) = (1, 36, 173). Chec: = 1.
15 4. SOLUTION TO EXERCISES 15 (4563, 1, 0) 4(981, 0, 1) = (639, 1, 4) (981, 0, 1) (639, 1, 4) = (342, 1, 5) (639, 1, 4) (342, 1, 5) = (297, 2, 9) (342, 1, 5) (297, 2, 9) = (45, 3, 14) (297, 2, 9) 6 (45, 3, 14) = (27, 20, 93) (45, 3, 14) (27, 20, 93) = (18, 23, 107) (27, 20, 93) (18, 23, 107) = (9, 43, 200) Chec: = 9. (451, 1, 0) 5(85, 0, 1) = (26, 1, 5) (85, 0, 1) 3 (26, 1, 5) = (7, 3, 16) (26, 1, 5) 3 (7, 3, 16) = (5, 10, 53) (7, 3, 16) (5, 10, 53) = (2, 13, 69) (5, 10, 53) 2 (2, 13, 69) = (1, 36, 191) (11111, 1, 0) 10(1111, 0, 1) = (1, 1, 10) (12345, 1, 0) 10(1234, 0, 1) = (5, 1, 10) (1234, 0, 1) 246 (5, 1, 10) = (4, 246, 2461) (5, 1, 10) (4, 246, 2461) = (1, 247, 2471) Chec: = These problems are very time consuming and unpleasant. The lesson is that with more mathematics life gets much easier. The answers given are obtained in the advanced fashion. 3.7 gcf(543, 112) = 1, lcm(543, 112) = / gcf(543, 112) = , gcf(4563, 981) = 9; lcm(4563, 981) = / gcf(4563, 981) = gcf(451, 85) = 1; = gcf(1111, 11111) = 1; lcm(1111, 11111) = = gcf(12345, 1234) = 1; lcm(12345, 1234) = = = = = = = = 211 is prime; 373 = 373 is prime; 453 = is composite; 565 = is composite; 463 = 463 is prime; 371 = 7 53 is composite 637 = is composite; 343 = 7 3 is composite; = is comp[osite; = 4026 = is composite = So q = 269 and r = = (1) 5 is a factor of x because 5 is a factor of 75y and 130.
16 16 3. NUMBER THEORY (2) 3 is not a factor of x because if it were, then 3 would be a factor of x and 75, hence of 75y and x, hence of 130 which is not true. (3) If y has a factor of 13, then x has a factor of 13. True. (4) If y is even, then x is also even. True. (5) If y is odd, then x is also odd. True. (6) If gcf(13, y) = 1, then 13 cannot be a factor of x. True. Assume to the contrary that 13 is a factor of x. Then 13 is a factor of x and 130 hence of 75y. Since gcf(13, y) = 1, we now that 13 is not a factor of y so it would have to be a factor of 75, which is false (1) odd odd = odd. This was done in class. (2) even even = even. Tae two even numbers x and y. Being even they are of the form x = 2x, y = 2y. Hence xy = 2x 2y = 4x y, so xy even has a factor 4 which is more than having a factor 2. (3) even odd = even. Let x be even and y be odd. Then x = 2x and y = 2y + 1. Hence xy = 2x y which is even. (4) odd + odd = even. True. (5) even + even = even. True Let a = b where b is some unnown positive integer such that Since gcf(2 5 19, b) = 1 any integer having only factors 2, 5, and 19 must be a factor of = So 16 : Yes, 80 Yes, 95 Yes, 361 = 19 2 No, 100 Yes, 76 = 4 19 Yes Loo for the complementary factors 378/1 = 378, 378/2 = 189, 378/3 = 126, 378/6 = 63, 378/7 = 54, 378/9 = 42, 378/14 = 27, 378/18 = 21. Hence the complete list is 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378.
Arithmetic, Algebra, Number Theory
Arithmetic, Algebra, Number Theory Peter Simon 21 April 2004 Types of Numbers Natural Numbers The counting numbers: 1, 2, 3,... Prime Number A natural number with exactly two factors: itself and 1. Examples:
More information5.1. Primes, Composites, and Tests for Divisibility
CHAPTER 5 Number Theory 5.1. Primes, Composites, and Tests for Divisibility Definition. A counting number with exactly two di erent factors is called a prime number or a prime. A counting number with more
More informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
More informationN= {1,2,3,4,5,6,7,8,9,10,11,...}
1.1: Integers and Order of Operations 1. Define the integers 2. Graph integers on a number line. 3. Using inequality symbols < and > 4. Find the absolute value of an integer 5. Perform operations with
More informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More informationDivisibility, Factors, and Multiples
Divisibility, Factors, and Multiples An Integer is said to have divisibility with another non-zero Integer if it can divide into the number and have a remainder of zero. Remember: Zero divided by any number
More informationCHAPTER 1 REAL NUMBERS KEY POINTS
CHAPTER 1 REAL NUMBERS 1. Euclid s division lemma : KEY POINTS For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b. 2. Euclid s division
More informationChapter 5: The Integers
c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition
More informationThe following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers:
Divisibility Euclid s algorithm The following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers: Divide the smaller number into the larger, and
More informationExam 2 Review Chapters 4-5
Math 365 Lecture Notes S. Nite 8/18/2012 Page 1 of 9 Integers and Number Theory Exam 2 Review Chapters 4-5 Divisibility Theorem 4-1 If d a, n I, then d (a n) Theorem 4-2 If d a, and d b, then d (a+b).
More informationREAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.
REAL NUMBERS Introduction Euclid s Division Algorithm Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Fundamental
More information2 Elementary number theory
2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},
More informationDirect Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24
Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using
More informationcse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska
cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska LECTURE 12 CHAPTER 4 NUMBER THEORY PART1: Divisibility PART 2: Primes PART 1: DIVISIBILITY Basic Definitions Definition Given m,n Z, we say
More informationINTEGERS. In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes.
INTEGERS PETER MAYR (MATH 2001, CU BOULDER) In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes. 1. Divisibility Definition. Let a, b
More informationand LCM (a, b, c) LCM ( a, b) LCM ( b, c) LCM ( a, c)
CHAPTER 1 Points to Remember : REAL NUMBERS 1. Euclid s division lemma : Given positive integers a and b, there exists whole numbers q and r satisfying a = bq + r, 0 r < b.. Euclid s division algorithm
More informationREVIEW Chapter 1 The Real Number System
REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }
More informationChapter 2. Divisibility. 2.1 Common Divisors
Chapter 2 Divisibility 2.1 Common Divisors Definition 2.1.1. Let a and b be integers. A common divisor of a and b is any integer that divides both a and b. Suppose that a and b are not both zero. By Proposition
More informationSlide 1 / 69. Slide 2 / 69. Slide 3 / 69. Whole Numbers. Table of Contents. Prime and Composite Numbers
Slide 1 / 69 Whole Numbers Table of Contents Slide 2 / 69 Prime and Composite Numbers Prime Factorization Common Factors Greatest Common Factor Relatively Prime Least Common Multiple Slide 3 / 69 Prime
More information1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11.
000 Chapter 1 Arithmetic in 1.1 The Division Algorithm Revisited 1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = 3. 2. (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11. 3. (a) q = 6, r =
More information1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11.
000 Chapter 1 Arithmetic in 1.1 The Division Algorithm Revisited 1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = 3. 2. (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11. 3. (a) q = 6, r =
More informationThe following techniques for methods of proofs are discussed in our text: - Vacuous proof - Trivial proof
Ch. 1.6 Introduction to Proofs The following techniques for methods of proofs are discussed in our text - Vacuous proof - Trivial proof - Direct proof - Indirect proof (our book calls this by contraposition)
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More information1 Overview and revision
MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction
More informationCHAPTER 4: EXPLORING Z
CHAPTER 4: EXPLORING Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction In this chapter we continue the study of the ring Z. We begin with absolute values. The absolute value function Z N is the identity
More informationNotes on arithmetic. 1. Representation in base B
Notes on arithmetic The Babylonians that is to say, the people that inhabited what is now southern Iraq for reasons not entirely clear to us, ued base 60 in scientific calculation. This offers us an excuse
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More informationNOTES ON SIMPLE NUMBER THEORY
NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,
More information4. Number Theory (Part 2)
4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.
More informationNumber Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some. Notation: b Fact: for all, b, c Z:
Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some z Z Notation: b Fact: for all, b, c Z:, 1, and 0 0 = 0 b and b c = c b and c = (b + c) b and b = ±b 1
More informationElementary Properties of the Integers
Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationNumber Theory and Divisibility
Number Theory and Divisibility Recall the Natural Numbers: N = {1, 2, 3, 4, 5, 6, } Any Natural Number can be expressed as the product of two or more Natural Numbers: 2 x 12 = 24 3 x 8 = 24 6 x 4 = 24
More informationCool Results on Primes
Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why
More informationExercises Exercises. 2. Determine whether each of these integers is prime. a) 21. b) 29. c) 71. d) 97. e) 111. f) 143. a) 19. b) 27. c) 93.
Exercises Exercises 1. Determine whether each of these integers is prime. a) 21 b) 29 c) 71 d) 97 e) 111 f) 143 2. Determine whether each of these integers is prime. a) 19 b) 27 c) 93 d) 101 e) 107 f)
More informationSection 3-4: Least Common Multiple and Greatest Common Factor
Section -: Fraction Terminology Identify the following as proper fractions, improper fractions, or mixed numbers:, proper fraction;,, improper fractions;, mixed number. Write the following in decimal notation:,,.
More information{ independent variable some property or restriction about independent variable } where the vertical line is read such that.
Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with
More informationMATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions
MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a
More informationNumber Theory and Graph Theory. Prime numbers and congruences.
1 Number Theory and Graph Theory Chapter 2 Prime numbers and congruences. By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 2 Module-1:Primes
More information(e) Commutativity: a b = b a. (f) Distributivity of times over plus: a (b + c) = a b + a c and (b + c) a = b a + c a.
Math 299 Midterm 2 Review Nov 4, 2013 Midterm Exam 2: Thu Nov 7, in Recitation class 5:00 6:20pm, Wells A-201. Topics 1. Methods of proof (can be combined) (a) Direct proof (b) Proof by cases (c) Proof
More informationChapter 7. Number Theory. 7.1 Prime Numbers
Chapter 7 Number Theory 7.1 Prime Numbers Any two integers can be multiplied together to produce a new integer. For example, we can multiply the numbers four and five together to produce twenty. In this
More informationQuestion 1: Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0?
Class IX - NCERT Maths Exercise (.) Question : Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0? q Solution : Consider the definition of a rational number.
More informationMATHEMATICS X l Let x = p q be a rational number, such l If p, q, r are any three positive integers, then, l that the prime factorisation of q is of t
CHAPTER 1 Real Numbers [N.C.E.R.T. Chapter 1] POINTS FOR QUICK REVISION l Euclid s Division Lemma: Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r
More informationa = qb + r where 0 r < b. Proof. We first prove this result under the additional assumption that b > 0 is a natural number. Let
2. Induction and the division algorithm The main method to prove results about the natural numbers is to use induction. We recall some of the details and at the same time present the material in a different
More informationPrime Factorization and GCF. In my own words
Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite
More information18 Divisibility. and 0 r < d. Lemma Let n,d Z with d 0. If n = qd+r = q d+r with 0 r,r < d, then q = q and r = r.
118 18. DIVISIBILITY 18 Divisibility Chapter V Theory of the Integers One of the oldest surviving mathematical texts is Euclid s Elements, a collection of 13 books. This book, dating back to several hundred
More informationA group of figures, representing a number, is called a numeral. Numbers are divided into the following types.
1. Number System Quantitative Aptitude deals mainly with the different topics in Arithmetic, which is the science which deals with the relations of numbers to one another. It includes all the methods that
More informationExample: This theorem is the easiest way to test an ideal (or an element) is prime. Z[x] (x)
Math 4010/5530 Factorization Theory January 2016 Let R be an integral domain. Recall that s, t R are called associates if they differ by a unit (i.e. there is some c R such that s = ct). Let R be a commutative
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More information1. Revision Description Reflect and Review Teasers Answers Recall of Rational Numbers:
1. Revision Description Reflect Review Teasers Answers Recall of Rational Numbers: A rational number is of the form, where p q are integers q 0. Addition or subtraction of rational numbers is possible
More informationMath Review. for the Quantitative Reasoning measure of the GRE General Test
Math Review for the Quantitative Reasoning measure of the GRE General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important for solving
More informationSolutions to Assignment 1
Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More information5: The Integers (An introduction to Number Theory)
c Oksana Shatalov, Spring 2017 1 5: The Integers (An introduction to Number Theory) The Well Ordering Principle: Every nonempty subset on Z + has a smallest element; that is, if S is a nonempty subset
More informationQ 1 Find the square root of 729. 6. Squares and Square Roots Q 2 Fill in the blank using the given pattern. 7 2 = 49 67 2 = 4489 667 2 = 444889 6667 2 = Q 3 Without adding find the sum of 1 + 3 + 5 + 7
More informationAssociative property
Addition Associative property Closure property Commutative property Composite number Natural numbers (counting numbers) Distributive property for multiplication over addition Divisibility Divisor Factor
More informationMath 110 FOUNDATIONS OF THE REAL NUMBER SYSTEM FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS
4-1Divisibility Divisibility Divisibility Rules Divisibility An integer is if it has a remainder of 0 when divided by 2; it is otherwise. We say that 3 divides 18, written, because the remainder is 0 when
More informationA number that can be written as, where p and q are integers and q Number.
RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.
More informationChapter V. Theory of the Integers. Mathematics is the queen of the sciences and number theory is the queen of mathematics. Carl Friedrich Gauss
Chapter V Theory of the Integers Mathematics is the queen of the sciences and number theory is the queen of mathematics. Carl Friedrich Gauss One of the oldest surviving mathematical texts is Euclid s
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationNumber Theory Solutions Packet
Number Theory Solutions Pacet 1 There exist two distinct positive integers, both of which are divisors of 10 10, with sum equal to 157 What are they? Solution Suppose 157 = x + y for x and y divisors of
More informationA field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:
Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under
More information3 The fundamentals: Algorithms, the integers, and matrices
3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationCHAPTER 8: EXPLORING R
CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed
More informationDivisibility. Chapter Divisors and Residues
Chapter 1 Divisibility Number theory is concerned with the properties of the integers. By the word integers we mean the counting numbers 1, 2, 3,..., together with their negatives and zero. Accordingly
More informationCh 3.2: Direct proofs
Math 299 Lectures 8 and 9: Chapter 3 0. Ch3.1 A trivial proof and a vacuous proof (Reading assignment) 1. Ch3.2 Direct proofs 2. Ch3.3 Proof by contrapositive 3. Ch3.4 Proof by cases 4. Ch3.5 Proof evaluations
More informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,
More informationArithmetic. Integers: Any positive or negative whole number including zero
Arithmetic Integers: Any positive or negative whole number including zero Rules of integer calculations: Adding Same signs add and keep sign Different signs subtract absolute values and keep the sign of
More informationChapter 1 A Survey of Divisibility 14
Chapter 1 A Survey of Divisibility 14 SECTION C Euclidean Algorithm By the end of this section you will be able to use properties of the greatest common divisor (gcd) obtain the gcd using the Euclidean
More informationSEVENTH EDITION and EXPANDED SEVENTH EDITION
SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 5-1 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to
More informationFinding Prime Factors
Section 3.2 PRE-ACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to
More informationMath 7 Notes Unit Two: Integers
Math 7 Notes Unit Two: Integers Syllabus Objective: 2.1 The student will solve problems using operations on positive and negative numbers, including rationals. Integers the set of whole numbers and their
More informationSummary: Divisibility and Factorization
Summary: Divisibility and Factorization One of the main subjects considered in this chapter is divisibility of integers, and in particular the definition of the greatest common divisor Recall that we have
More informationFoundations Revision Notes
oundations Revision Notes hese notes are designed as an aid not a substitute for revision. A lot of proofs have not been included because you should have them in your notes, should you need them. Also,
More informationDaily Skill Builders:
Daily Skill Builders: Pre-Algebra By WENDI SILVANO COPYRIGHT 2008 Mark Twain Media, Inc. ISBN 978-1-58037-445-3 Printing No. CD-404086 Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing
More informationKnow the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.
The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring
More informationChapter 14: Divisibility and factorization
Chapter 14: Divisibility and factorization Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter
More informationThe set of integers will be denoted by Z = {, -3, -2, -1, 0, 1, 2, 3, 4, }
Integers and Division 1 The Integers and Division This area of discrete mathematics belongs to the area of Number Theory. Some applications of the concepts in this section include generating pseudorandom
More informationCSE 215: Foundations of Computer Science Recitation Exercises Set #5 Stony Brook University. Name: ID#: Section #: Score: / 4
CSE 215: Foundations of Computer Science Recitation Exercises Set #5 Stony Brook University Name: ID#: Section #: Score: / 4 Unit 10: Proofs by Contradiction and Contraposition 1. Prove the following statement
More informationBeautiful Mathematics
Beautiful Mathematics 1. Principle of Mathematical Induction The set of natural numbers is the set of positive integers {1, 2, 3,... } and is denoted by N. The Principle of Mathematical Induction is a
More informationMath1a Set 1 Solutions
Math1a Set 1 Solutions October 15, 2018 Problem 1. (a) For all x, y, z Z we have (i) x x since x x = 0 is a multiple of 7. (ii) If x y then there is a k Z such that x y = 7k. So, y x = (x y) = 7k is also
More informationSlides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking
More informationMATH10040 Chapter 1: Integers and divisibility
MATH10040 Chapter 1: Integers and divisibility Recall the basic definition: 1. Divisibilty Definition 1.1. If a, b Z, we say that b divides a, or that a is a multiple of b and we write b a if there is
More informationExecutive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics:
Executive Assessment Math Review Although the following provides a review of some of the mathematical concepts of arithmetic and algebra, it is not intended to be a textbook. You should use this chapter
More informationMasters Tuition Center
1 REAL NUMBERS Exercise 1.1 Q.1. Use Euclid s division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 Solution. (i) In 135 and 225, 225 is larger integer. Using Euclid
More informationThis is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time.
8 Modular Arithmetic We introduce an operator mod. Let d be a positive integer. For c a nonnegative integer, the value c mod d is the remainder when c is divided by d. For example, c mod d = 0 if and only
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationCISC-102 Fall 2017 Week 6
Week 6 page 1! of! 15 CISC-102 Fall 2017 Week 6 We will see two different, yet similar, proofs that there are infinitely many prime numbers. One proof would surely suffice. However, seeing two different
More informationChapter 1. Greatest common divisor. 1.1 The division theorem. In the beginning, there are the natural numbers 0, 1, 2, 3, 4,...,
Chapter 1 Greatest common divisor 1.1 The division theorem In the beginning, there are the natural numbers 0, 1, 2, 3, 4,..., which constitute the set N. Addition and multiplication are binary operations
More informationContest Number Theory
Contest Number Theory Andre Kessler December 7, 2008 Introduction Number theory is one of the core subject areas of mathematics. It can be somewhat loosely defined as the study of the integers. Unfortunately,
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 8, 2013 Robb T. Koether (Hampden-Sydney College) Direct Proof Divisibility Fri, Feb 8, 2013 1 / 20 1 Divisibility
More information1. multiplication is commutative and associative;
Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. College Algebra for STEM
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics College Algebra for STEM Marcel B. Finan c All Rights Reserved 2015 Edition To my children Amin & Nadia Preface From
More information