5.1 Number Theory

Euclidean Algorithm GCD

- let a and b be any two natural numbers with a ≥ b - divide a by b to get a remainder r - If r = 0 then b is the GCD - If r > ​0 divide b by to get a remainder s - If s = 0 the r is the GCD - If s > 0 divide r by s to get a remainder of t - continue until you get a remainder of 0 - the last nonzero remainder is GCD

A perfect number is a natural number that is equal to the sum of its​ factors, excluding the number itself. Determine whether or not the given number is perfect. 12

12 is NOT a perfect number

Brenda cuts her grass every 15 days and trims her shrubs every 24 days. If Brenda cut her grass and trimmed her shrubs on June 1 how many days will it be before she cuts her grass and trims her shrubs on the same day again?

120 days * first find the factorization for 15 * next find the factorization for 24 2^3 x 3 x 5 = 120

Two numbers with a greatest common divisor of 1 are said to be relatively prime. For​ example, the numbers 9 and 14 are relatively​ prime, since their GCD is 1. 134, 249

134 and 249 are relatively prime because their GCD is 1

Two numbers with a greatest common divisor of 1 are said to be relatively prime. For​ example, the numbers 9 and 14 are relatively​ prime, since their GCD is 1. 14, 15

14 and 54 are not relatively prime because their GCD is 2

____ is the least common multiple of 30 and 36

180

Find the prime factor of 506

2 x 11 x 23

Jen collects toy sport cars. She has 1568 red cars and 448 green cars. She wants to line up the cars in groups so that each group has the same number of cars and contains only red cars or only green cars. what is the largest number of cars that she can have?

224 * factor 1568 2^5 x 7^2 * factor 448 2^6 x 7 * find the smallest exponents: 2^5 x 7 * complete the math = 224

Write 1200 as a product of primes.

2^4 x 3 x 5^2

___ is the factor for both 30 and 36, but NOT the greatest common factor

3

Find the prime factorization of 99

3 x 3 x 11 * first find the smallest prime # of 99 answer: 3 * next divide 99 by 3 which = 33 * find the smallest prime # of 33 answer: 3 * divide 33 by 3 which = 11 * 11 cannot be factored hence the answer above

A perfect number is a natural number that is equal to the sum of its​ factors, excluding the number itself. Determine whether or not the given number is perfect. 32

32 is NOT a perfect number * list factors: 1, 2, 4, 8, 16 * find the sum of factors = 31

____ is a multiple of both 30 and 36, but is NOT the least common multiple

360

Find the prime factorization of 1287

3^2 x 11 x 13

Two numbers with a greatest common divisor of 1 are said to be relatively prime. For​ example, the numbers 9 and 14 are relatively​ prime, since their GCD is 1. 55, 44

55 and 44 are not relatively prime because their GCD is 11 55 = 5 x 11 44 = 2^2 x 11

___ is the greatest common factor of 30 and 36

6

A perfect number is a natural number that is equal to the sum of its​ factors, excluding the number itself. Determine whether or not the given number is perfect. 80

80 is NOT a perfect number

Which of the following numbers is divisible by 2​, 3​, 4​, 6​, and 7​? a) 83 b) 86 c) 84 d) 85

84

Two numbers with a greatest common divisor of 1 are said to be relatively prime. For​ example, the numbers 9 and 14 are relatively​ prime, since their GCD is 1. Determine whether the following pairs of numbers are relatively prime. 9, 8

9 and 8 are relatively prime because their GCD is 1. 9 = 3^3 8 = 2^3

True /False: "6 is a multiple of 30"

False * 30 is a multiple of 6

True /False: "9 is a multiple of 72"

False * 72 is a multiple of 9

Use the Euclidean algorithm to find the GCD. 20, 75

The GCD of 20 and 75 = 5 75/20 = 3 r 15 20/ 15 = 1 r 5 5/1 = 0

Use the Euclidean algorithm to find the GCD 32, 160

The GCD of 32 and 160 = 32 160/ 32 = 5 r 0

True /False: "7 is a factor of 14"

True

True /False: If a number is not divisible by 5, then it is not divisible by 10"

True * A number is divisible by 5 IF the number ends in 5 OR 0. A number is divisible by 10 IF the number ends in 0.

A supposition or hypothesis that has not been proved or disproved is known as

a conjuncture

Jerrett is setting up chairs for his school band concert. He needs to put 270 chairs on the gymnasium floor in rows of equal​ size, and there must be at least 2 chairs in each row and at least 2 rows. List the number of rows and the number of chairs in each row that are possible.

a) 2 rows by 135 chairs b) 18 rows by 15 chairs c) 5 rows by 54 chairs e) 3 rows by 90 chairs f) 135 rows by 2 chairs g) 45 rows by 6 chairs h) 6 rows by 45 chairs i) 90 rows by 3 chairs k) 15 rows by 18 chairs l) 27 rows by 10 chairs s m) 54 rows by 5 chairs n) 30 rows by 9 chairs r) 10 rows by 27 chairs t) 9 rows by 30 chairs

A natural number that is divisible by a number other than itself and 1 is known as a

composite number

The smallest natural number that is divisible (without a remainder) by each number in a set of numbers is known as the least common ______

multiple of the set of number

The study of numbers and their properties is known as

number theory

Factors of a natural number are ___ than multiples

smaller

The GCF of two numbers is always ____ than the LCM of the two numbers.

smaller

If a is divisible by​ b, then a ÷ b has a remainder of

zero

The following are prime numbers up to​ 100

​2, 3,​ 5, 7,​ 11, 13,​ 17, 19,​ 23, 29,​ 31, 37,​ 41, 43,​ 47, 53,​ 59, 61,​ 67, 71,​ 73, 79,​ 83, 89, 97