Real Number. Euclid s division algorithm is based on the above lemma.


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1 1 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius Basic Concepts 1. 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. Here a and b are called dividend and divisor respectively, q is called quotient and r the remainder. Euclid s division algorithm is based on the above lemma. 2. Method to find HCF of two positive integers a and b (a > b) using Euclid s division algorithm Step1. By Euclid s algorithm, find q and r where a = bq + r, 0 r < b. Step2. If r = 0, the HCF is b. if r 0, apply the algorithm to b and r. Step3. Continue the process till the remainder is zero. The divisor at this stage is the required HCF. 3. The Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of primes and this factorization is unique, apart from the order in which the prime factors occurs. 4. If a prime number p divides a 2, then p also divides a where a is a positive integer. 5. A rational number with terminating decimal expansion can always be expressed in the form q p where p and q are coprime and the prime factorization of q is of the form 2 m.5 n where m and n are nonnegative integers and vice versa. 6. If a rational number is of the form q p and q is not of the form 2 m.5 n, then its decimal representation is nonterminating but repeating. 7. A number is called irrational if it cannot be expressed in the form q p where p and q are integers are q 0. Example 1: Use Euclid s division algorithm to find the HCF of 135 and 225. Sol. HCF of 135 and 225; Applying the Euclid s lemma to 225 and 135, (where 225 > 135), we get 225 = (135 1) + 90, since 90 0, therefore, applying the Euclid s lemma to 135 and 90, we have 135 = (90 1) + 45 But 45 = Applying Euclid s lemma to 90 and 45, we get 90 = (45 2) + 0 Here, r = 0, so our procedure stops. Since, the divisor at the last step is 45. HCF of 225 and 135 is 45. [1]
2 Example 2: Find the H.C.F. and L.C.M. of and by using the Fundamental Theorem of Arithmetic. Sol. The prime factors of and are given below : = = H.C.F. (25152, 12156) = Product of the smallest power of each common prime factors in the number = = 12. We know that H.C.F. (a, b) L.C.M. (a, b) = a b H.C.F (25152, 12156) L.C.M. (25152, 12156) = L.C.M. (25152, 12156) = L.C.M. (25152, 12156) = 12 = = EXERCISE Use Euclid s division algorithm to find the HCF of : (i) 4052 and (ii) 196 and (iii) 867 and Using Euclid s Division Algorithm find the H.C.F. of 36575, 2223 and Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) Find the LCM and HCF of 6 and 20 by the prime factorization method. 5. Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM. 6. Find the HCF and LCM of 6, 72 and 120, using the prime factorization method. 7. Find the LCM and HCF of the following pairs of integers and verify that LCM HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and Find the LCM and HCF of the following integers by applying the prime factorization method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and Find the L.C.M and H.C.F. of the following integers by applying the prime factorization method. (i) 275, 225, 175 (ii) 765, 510, 408 (iii) 19, 13, Find the L.C.M. and H.C.F. of the following integers by applying the prime factorization method. (i) 2800, 1680, 1200 (ii) 1335, 890, 534 (iii) 392, 343, Given that HCF (306, 657) = 9, find LCM (306, 657). 12. Given HCF of (16, 100) = 4, write LCM of (16, 100) 13. Given LCM of (26, 169) = 338, what is HCF of (26, 169)? [2]
3 Example 3 : Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. Sol. Let a be any positive integer and b = 3 a = 3q + r, where q 0 and 0 r < 3 a = 3q or 3q + 1 or 3q + 2 Therefore, every number can be represented as these three forms. There are three cases. Case 1: When a = 3q, a 3 = (3q) 3 = 27q 3 = 9(3q) 3 = 9m, Where m is an integer such that m = 3q 3 Case 2: When a = 3q + 1, a 3 = (3q +1) 3 a 3 = 27q q 2 + 9q + 1 a 3 = 9(3q 3 + 3q 2 + q) + 1 a 3 = 9m + 1 Where m is an integer such that m = (3q 3 + 3q 2 + q) Case 3: When a = 3q + 2, a 3 = (3q +2) 3 a 3 = 27q q q + 8 a 3 = 9(3q 3 + 6q 2 + 4q) + 8 a 3 = 9m + 8 Where m is an integer such that m = (3q 3 + 6q 2 + 4q) Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8. EXERCISE Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. 2. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. 3. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. 4. Use Euclid s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 5. Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q. 6. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q. [3]
4 Example 4: Check whether 6 n can end with the digit 0 for any natural number n. Sol. If any digit has last digit 10 that means it is divisible by 10 and the factors of 10 = 2 5. So value 6 n should be divisible by 2 and 5 both 6 n is divisible by 2 but not divisible by 5 So it can not end with 0. Example 5: Explain why and are composite numbers. Sol Taking 13 common, we get 13 (7 x ) 13( ) 13 (78) It is product of two numbers and both numbers are more than 1 so it is a composite number Taking 5 common, we get 5( ) 5( ) 5(1009) It is product of two numbers and both numbers are more than 1 so it is a composite number. EXERCISE A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of barfis that can be placed in each stack for this purpose? 2. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? 3. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? 4. In a seminar, the number of participants in Hindi, English and are 60, 84 and 108, respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject. 5. Three sets of English, Hindi and books have to be stacked in such a way that all the books are stored topic wise and he height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and books goats, 140 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. Can you tell how many animals went in each trip? [4]
5 7. 15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxed with the same number of pastries and biscuit packets in each. How many biscuit packets and how many pastries will each box contain? 8. A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size inches of the tile required that has to be cut and how many such tiles are required? 9. What is the smallest number that, when divided by 35, 56 and 91 leaves remainders of 7 in each case? 10. In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps? 11. Determine the number nearest to but greater than which is exactly divisible by each of 8, 15 and Consider the numbers 4 n, where n is a natural number. Check whether there is any value of n for which 4 n ends with the digit zero. 13. Check whether 5 n can end with the digit 0 for any n N. 14. Check whether 8 n can end with the digit 0 for any n N. 15. Check whether 12 n can end with the digit 5 for any n N. 16. Check whether 2 n can end with digit 6 for any n N. 17. Check whether 3 n can end with the digit 1, n 1, n N. Example 6 : Prove that Sol. Let n be a rational number. p n [p and q are coprime] q 2 p n [Squaring both sides] 2 q p 2 = nq 2 n divides p 2 n divides p n is not a rational number, if n is not a perfect square. Let p = nm p 2 = n 2 m 2 [Squaring both sides] Putting the value of p 2 in (1), we get.(1).(2) n 2 m 2 = nq 2 q 2 = nm 2 n divides q 2 n divides q.(3) From (2), n divides p and from (3) n divides q. It means n is a common factor of both p and q. This contradicts the assumption that p and q are coprime. So, our supposition is wrong. Hence, n cannot be a rational number. [5]
6 Example 7 : Prove that is irrational. Sol. Let take that is a rational number. So we can write this number as = a/b Here a and b are two co prime number and b is not equal to 0 Subtract 3 both sides we get 2 5 = a/b = (a3b)/b Now divide by 2, we get 5 = (a3b)/2b Here a and b are integer so (a3b)/2b is a rational number so 5 should be a rational number But 5 is a irrational number so it contradicts. Hence, is a irrational number. Question based on proof of irrational number EXERCISE If a prime number p divides a 2, then p also divides a where a is a positive integer. 2. A number is called irrational if it cannot be expressed in the form q p where p and q are integers are q Show that 2 is an irrational number. 4. Show that 3 is not a rational number. 5. Show that 5 3 is irrational. 6. Show that is an irrational number. 7. Show that is not a rational number. 8. Show that is not a rational number. 9. Show that 7 3 is an irrational number. 10. Show that 1 is irrational Show that sum and product of two irrational numbers and 7 5 are rational numbers. 12. Prove that 3 2 is irrational. 13. Prove that 5 is an irrational number. 14. Prove that 5 3 is an irrational number. 15. Prove that 2 5 is irrational. [6]
7 A rational number with terminating decimal expansion can always be expressed in the form q p where p and q are coprime and the prime factorization of q is of the form 2 m.5 n are nonnegative integers and vice versa. where m and n If a rational number is of the form q p and q is not of the form 2 m.5 n, then its decimal representation is nonterminating but repeating. EXERCISE 1.5 Question based on Terminating and nonterminating decimals 1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a nonterminating repeating decimal expansion: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) Without actually performing long division sate whether the following rational numbers have a terminating decimal or nonterminating recurring decimal expression : 111 (i) 125 (ii) (iii). 455 (iv) Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a nonterminating repeating decimal expansion. (i) 23 (ii) 125 (iii) 35 (iv) (v) Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2 m 5 n, where m, n are nonnegative integers: (i) 3 (ii) (iii) 7 (iv) (v) What can you say about the prime factorizations of the denominators of the following rationals: (i) (ii) [7]
8 Exercise 1.1 ANSWER (1) (i) H.C.F. = 4 (ii) 196 (iii) 51 (2) 1 (3) (i) (ii) (iii) (iv) (v) (4) H.C.F. = 2, L.C.M. = 60 (5) H.C.F. = 2 2, L.C.M. = (6) H.C.F. = 2, L.C.M. = (7) (i) L.C.M. = 182, H.C.F. = 13 (ii) L.C.M. = 23460, H.C.F. = 2 (iii) L.C.M. = 3024, H.C.F. = 6 (8) (i) L.C.M. = 420, H.C.F. = 3 (ii) L.C.M. = 11339, H.C.F. = 1 (iii) L.C.M. = 1800, H.C.F. = 1 (9) (i) H.C.F. = 25, L.C.M. = (ii) H.C.F. = 51, L.C.M (iii) H.C.F. = 1, L.C.M. = 1729 (10) (i) H.C.F. = 80, L.C.M. = 8400 (ii) H.C.F. = 89, L.C.M. = 2670 (iii) H.C.F. = 49, L.C.M. = 8232 (11) (12) L.C.M. = 400 (13) H.C.F. = 13 Exercise 1.2 (1) H.C.F. = 10, So, the Sweetseller can make stack of 10 for both kind of barfi. (2) 8 columns (3) 36 minutes (4) 12 room (5) 48 stacked (6) 35 Animal Exercise 1.5 (1) (i) terminating (ii) Terminating (iii) non terminating repeating (iv) Terminating (v) Non terminating repeating (vi) Terminating (vii) non terminating repeating (viii) Terminating (2) (i) Terminating (ii) Terminating (iii) non terminating repeating (iv) non terminating repeating (3) (i) Terminating (ii) non terminating repeating (iii) Terminating (iv) non terminating repeating (v) non terminating repeating (4) (i) Terminating (ii) Terminating (iii) Terminating (iv) Terminating (v) Terminating (5) (i) The denominators is of the form of 2 m 5 n. (ii) The denominator is not in the form of 2 m 5 n. [8]
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