number theory and...

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

Divisibility Ex. Divisors of 24

1 | 24 because 24 ÷ 1 = 24 with no remainder








5 ∤ 24 because 24 ÷ 5 gives us a nonzero remainder

7 ∤ 24 because 24 ÷ 5 gives us a nonzero remainder

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 = 22 x 52 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 = 22 x 3 16 = 24

gcd(12,16) = 22 = 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 25 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 = 22 x 3 and 16 = 24

Prime 2 has highest exponent 4

Prime 3 has highest exponent 1

Therefore lcm(16,12) = 24 x 3 = 48


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?


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 Problems6

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

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