Lesson 1Multiples and Factors

At times you will be asked to determine the factors or multiples of various numbers in math problems in school and on the Terra Nova test. Many students confuse the two, so it is especially important to understand the differences between them.

Factors

• The factors of a number are those integers that divide a number without a remainder. All numbers have a least two factors – one and the number itself. Numbers that have only those factors are called prime numbers.

Example 1

• What are the factors of 81?

• Strategy: Find the pairs of numbers you can multiply together to get a product of 81.

• Step 1: Always start with one and the number itself. 1 x 81 = 81

• Step 2: Using the factoring rules 2, 3, 4, 5, etc., think of the other pairs of numbers that result in the product of 81.

• 3 x 27 and 9 x 9

Solution

• The factors of 81 are 1, 3, 9, 27, and 81. (Note: You only list 9 once, since it multiplies itself to get a result of 81)

Prime Factors

• The prime factors of a number are the prime numbers that can be multiplied together to yield the number as a product. All numbers can be expressed in terms of their prime factorization.

Example 2• What is the prime factorization of 36?

• Strategy: Pick a prime number that 36 can be divided by (use the factor rules to make your selection), continue this until the number left is prime.

• Step 1: 36 can be divided by 2, leaving 18.

• Step 2: 18/2 = 9; 9/3 = 3; 3 is prime.

2 36

2 18

3 9

3

• Write the prime factors, use exponents if asked

• 36 = 2 x 2 x 3 x 3

• 36 = 22 x 32

Solution

• The prime factorization of 36 is 2 x 2 x 3 x 3 or 22 x 32.

Greatest Common Factor (GCF)

• The greatest common factor is always between 2 or more numbers. It is the greatest number that is a factor to them all. You use the GCF when you are looking to reduce a fraction.

Example 3

• What is the GCF of 16 and 40?

• Strategy: Find the prime factorization of each number first.

• Step 1: The prime factorization of 16 is 2 x 2 x 2 x 2.

• Step 2: The prime factorization of 40 is 2 x 2 x 2 x 5.

• Step 3: Intersect (match up) the sets of factors.

• 2 x 2 x 2 is the match of the two numbers, therefore 8 is the greatest matching factor.

• Step 4: The GCF of 16 and 40 is 8.

Solution

• The GCF of 16 and 40 is 8.

Multiples• Finding the multiples of a number is as

easy as counting by that number. So, the multiples of 16 are 16, 32, 48, 64, etc. Another way you can express multiples is by multiplying that number by the counting numbers, such as 16 x 1, 16 x 2, 16 x 3, 16 x 4, etc.

• Because of the nature of multiples, they will go on forever. There are an infinite number of multiples of a given number. However, there are a finite number of factors for any given number.

Example 4

• What are the first 6 multiples of 12?

• 1 x 12 = 12, 2 x 12 = 24, 3 x 12 = 36,

4 x 12 = 48, 5 x 12 = 60, 6 x 12 = 72

Solution

• The first 6 multiples of 12 are 12, 24, 36, 48, 60, and 72.

Least Common Multiple (LCM)

• The least common multiple (LCM) of two or more numbers is the smallest of the numbers that is a multiple of all of them. You have been finding the LCM whenever you are looking for a common denominator among fractions.

Example 5

• What is the LCM of 8 and 12?

• Strategy: Find the prime factorization of each number first.

• Step 1: The prime factorization of 8 is 23.

• Step 2: The prime factorization of 12 is 22 x 3.

• Step 3: Intersect the factors (match the factors). Look for the lowest exponent of each different factor. 22 is the lowest of the twos and 3 is the lowest of the threes. 22 x 3 or 4 x 3 is the LCM.

Solution.

• The LCM of 8 and 12 is 24.