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 Factors of 24 Factors of 24 Factors of 24 Factors of 24: Natural Numbers that multiply to give us 24
Number Theory and Divisibility Factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24 1 x 24 = 24 2 x 12 = 24 3 x 8 = 24 4 x 6 = 24 6 x 4 = 24 8 x 3 = 24 12 x 2 = 24 24 x 1 = 24
Divisibility a is divisible by b, if the operation of dividing a by b leaves a remainder of 0 b a b is a divisor of a b divides a The following are Equivalent Statements: 24 is divisible by 8 8 is a divisor of 24 8 divides 24 8 24
Ex. Divisors of 24 Divisibility 1 24 because 24 1 = 24 with no remainder 2 24 because 24 2 = 12 with no remainder 3 24 because 24 3 = 8 with no remainder 4 24 because 24 4 = 6 with no remainder 6 24 because 24 6 = 4 with no remainder 8 24 because 24 8 = 3 with no remainder 12 24 because 24 12 = 2 with no remainder 24 24 because 24 24 = 1 with no remainder 5 24 because 24 5 gives us a nonzero remainder 7 24 because 24 5 gives us a nonzero remainder
Divisibility True or False? 4 16 True since 16 4 = 4 2 15 False since we get a remainder 7 14 True since 14 7 = 2 3 15 True since 15 3 = 5 2 208 True since 208 2 = 104
Divisibility Rules
Divisibility True or False? Use the Rules of Divisibility 8 48,324 False, last 3 digits 324 not divisible by 8 6 48,324 True, divisible by 2 and 3 4 48,324 False, 4 does divide 48,324 since 4 24
Prime Factorization Prime Number: a Natural Number greater than 1 that has only itself and 1 as factors. Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, Composite Number: a Natural Number greater than 1 that is divisible by a number other than 1 and itself. Ex. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, Note: 1 is neither prime nor composite by definition
Prime Factorization Every Composite Number can be expressed as the product of prime numbers. Expressing a number in this form is called Prime Factorization. Ex. 45 = 3 x 3 x 5 18 = 2 x 3 x 3 42 = 7 x 3 x 2
Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of prime numbers in one and only one way. (Order does not matter) This means Prime Factorization is unique to every composite number.
Factor Tree A method to find the prime factorization of a composite number Step 1) Select two numbers (other than 1) that are factors of your number Step 2) Repeat this process for the factors that are composite numbers until you only have primes left
Factor Tree Ex. Find the prime factorization of 700 700 = 2 2 x 5 2 x 7 Ex. Find the prime factorization of 120
Greatest Common Divisor gcd(a,b) = the largest number that is a divisor (factor) of a and b. If gcd(a,b) = 1, then a and b are Relatively Prime Finding GCD using Prime Factorization 1) Write prime factorization of each number 2) Select each prime factor with the smallest exponent that is common to each prime factorizations 3) The GCD is the product of the numbers in part 2
Greatest Common Divisor Find the GCD of 16 and 12 12 16 4 3 4 4 2 2 2 2 2 2 12 = 2 2 x 3 16 = 2 4 gcd(12,16) = 2 2 = 4
Greatest Common Divisor Find the GCD of 40 and 24
Greatest Common Divisor For an intramural league, you need to divide 192 men and 288 women into all-male and all-female teams so that each team has the same number of people. What is the largest number of people that can be placed on a team?
Greatest Common Divisor 192 men divided into teams: number of men per team is a divisor of 192 288 women divided into teams: number of women per team is a divisor of 288 The number of people per team must be the same for both men and women. We are looking for the largest number for this occur, i.e. the largest number that divides both 192 and 288 without a remainder
Greatest Common Divisor We want to find the GCD of 192 and 288 192 288 96 2 2 144 32 3 12 12 4 8 4 3 4 3 2 2 2 2 2 2 2 2 2 Therefore, the greatest number of people that can be placed into teams is 2 5 x 3 = 96
Least Common Multiple lcm(a,b) = smallest Natural Number that is divisible by a and b Find LCM by making a list of multiples of each number Ex. Find lcm(15,20) Multiples of 15: {15, 30, 45, 60, 75, 90, 105, 120, } Multiples of 20: {20, 40, 60, 80, 100, 120, 140, 160, } 60 and 120 are common multiples. The least common multiple is 60.
Least Common Multiple Method 2: Finding LCM using Prime Factorization 1) Write the prime factorization of each number 2) Select every prime raised to the greatest power that it occurs 3) The product of numbers in Step 2 is the LCM
Least Common Multiple Find the LCM of 16 and 12 Recall that: 12 = 2 2 x 3 and 16 = 2 4 Prime 2 has highest exponent 4 Prime 3 has highest exponent 1 Therefore lcm(16,12) = 2 4 x 3 = 48
Least Common Multiple A movie theater runs two documentary films continuously. One documentary runs for 40 minutes and a second documentary runs for 60 minutes. Both movies begin at 3PM. When will the movies begin again at the same time?
Least Common Multiple The documentaries repeat and we want to find when they both start at the same time. Common multiples of 40 and 60 will give us the minutes after 3PM when both movies start at the same time. We want to find the LCM, which is the first time they start together again. lcm(40,60) = 120 120 minutes = 2 hours 2 hours from 3PM is 5PM
Practice Problems Page 256 Divisible? #1, 7, 11, 17, 21 Prime Factorization: #25, 27, 31, 35, 39, 41 GCD: #45, 49, 51, 53, 55 LCM: #57, 59, 61, 65, 67 Applications: #91, 98