Factoring Integers

Rule 9: Factoring Integers

To factor an integer, simply break the integer down into a group of numbers whose product equals the original number. Factors are separated by multiplication signs. Note that the number 1 is the factor of every number. All factors of a number can be divided evenly into that number.

Example 1: Factor the number 3.

Answer:
Since 3 x 1 = 3, the factors of 3 are 3 and 1.

Example 2: Factor the number 10.

Answer:
Since 10 can be written as 5 x 2 x 1, the factors of 10 are 10, 5, 2, and 1. The number 10 can be divided by 10, 5, 2, and 1.

Example 3: Factor the number 18.

Answer:
The number 18 can be written as 18 x 1 or 9 x 2 or 6 x 3 or 3 x 3 x 2. Since 18 can be divided by 18, 9, 6, 3, 2, and 1, then 18, 9, 6, 3, 2, and 1 are factors of 18.

Example 4: Factor the number 24.

Answer:
The number 24 can be written as 24 x 1 or 12 x 2 or 8 x 3 or 4 x 6 or 2 x 2 x 2 x 3. Since 24 can be divided by 24, 12, 8, 6, 4, 3, 2, and 1, then 24, 12, 8, 6, 4, 3, 2, and 1 are factors of 24.

Example 5: Factor the number 105.

Answer:
The number 105 can be written as 105 x 1 or 21 x 5 or 3 x 7 x 5 or 15 x 7 or 35 x 3. Since 105 can be divided by 105, 35, 21, 15, 7, 5, 3, and 1, then 105, 35, 21, 15, 7, 5, 3, and 1 are factors of 105.

Example 6: Factor the number 1200 completely.

Answer:
This instruction means to factor 1200 into a set of prime factors (factors that cannot again be factored). The number 1200 can be written as 1200 x 1 or 100 x 12. Note the 100 can again be factored to 10 x 10 and the 12 can be factored to 6 x 2. So now you have 1200 = 100 x 12 = 10 x 10 x 6 x 2. This factored set can again be factored to (2 x 5) x (2 x 5) x (2 x 3) x 2 x 1. The number 1200 is factored completely as 5 x 5 x 3 x 2 x 2 x 2 x 2 x 1.

Work the following problems for practice with this skill.

Problem 1: Factor 15 completely.

Problem 2: Factor 300 completely.

Problem 3: Factor 4000 completely.

Problem 4: Factor -3 completely.

Problem 5: Is 3 a factor of 10? Why?

Problem 6: Is 7 a factor of 21? Why?

Problem 7: Is 4 a factor of 87? Why?