5.1 Divisibility

Natural Numbers

• The set of natural numbers or counting numbers is {1,2,3,4,5,6,…}

Factors

• The factors of a number are numbers that are multiplied together to equal that number.

• Example: What are the factors of 12?

So the factors of 12 are 1, 2, 3, 4, 6, & 12. If 12 is divided by any of its factors the remainder is zero.

1 12 12

2 6 12

3 4 12

Divisibility

• We say a is divisible by b if dividing a by b leaves a remainder of 0.

• We say that b is a divisor of a.• Example:

Since with no remainder we say that

24 is divisible by 8

8 divides 24

8 is a divisor of 24

We write 8|24

24 8 3

Factors and divisibility

• Factors and divisors are the same.

• For example:

8 is a factor and divisor of 16 since

2 8 16 and 16 8=2

Review the rules of divisibility p. 144

Examples• 4,681,396 is divisible by 2 since 6 is even• 5,931,471 is divisible by 3 since 5 + 9 + 3 + 1 + 4 + 7

+ 1 = 30 is divisible by 3• 4,865,924 is divisible by 4 since 4 | 24• 954 is divisible by 6 since 2 | 954 and

3 | 954• 30,385 is divisible by 5 since it ends in 5 or 0• 593,777,832 is divisible by 8 since the 8|832• 543,186 is divisible by 9 since 5 + 4 + 3 + 1 + 8 + 6=

27 is divisible by 9• 35,780 is divisible by 10 since it ends in 0• 614,608,176 is divisible by 12 since 3 and 4 divide it

Prime Numbers

• A prime number is a number greater than 1 with only 2 divisors or factors; 1 and itself.

• Example: 2, 3, 5, 7, 11, 13, 17, …

• Activity: Sieve of Eratosthenes

Composite Numbers

• A composite number is a number > 1 with a factor other than 1 and itself.

• For example: 4, 6, 8, 9, 10, 12, 14, 15,…

Prime Factorization

• The prime factorization of a number is expressing it as a product of its prime factors.

Factor Trees

• We can show prime factorization using a factor tree:

340

34 10

2 17 2 5

So 340 = 22 2 5 17 2 5 17

Write a factor tree for the following numbers

• 700

• 180

• 510

Greatest Common Factor

• The Greatest Common Factor or GCF is the greatest divisor of all the numbers.

• To find:1. Write the prime factorization of each

number2. Select factors that are common to each3. Take the smallest power of each of the

factors selected4. Multiply

Examples

• Find the GCF of 225 and 825

• Find the GCF of 72 and 120

Relatively Prime

• If two numbers share no common factors other than one then they are called relatively prime.

• Example: 35 and 12 are relatively prime since they share no common factors other than 1

Least Common Multiple

• The Least Common Multiple is the smallest number divisible by all of the numbers.

• One way to find the LCM is to list all multiples of each number and circle the smallest common one

• Example: To find the LCM of 15 and 20Multiples of 15: 15, 30, 45, 60, 75,…Multiples of 20: 20, 40, 60, 80,…The LCM of 15 and 20 is 60.

2nd Way to Find LCM

1. Write prime factorization of each number

2. Select every factor

3. Take the highest power of each factor

4. Multiply

Example

• Find the LCM of 18 and 30

• Find the LCM of 144 and 300

• Find the LCM of 60 and 108

HW: p. 200/1-10,25-68