Once you can find the greatest common factor of two numbers, you can write their sum as one product. This is the distributive property used backward, and it is the same move you will soon make with letters.
Key idea
To factor a sum, pull out the GCF and divide each number by it: $36 + 8 = 4(9 + 2)$, because $36 = 4 \times 9$ and $8 = 4 \times 2$.
Why it works
Picture two rectangles with the same height, 4. One has area 36 (4 by 9), the other has area 8 (4 by 2). Push them together and you get one rectangle 4 tall and $9 + 2$ long. Its area is $4(9 + 2) = 44$, which is $36 + 8$. That is the distributive property: $4(9 + 2) = 4 \times 9 + 4 \times 2$.
Worked example: Pull out the GCF
Write $24 + 60$ as a product using the GCF.
- The GCF of 24 and 60 is 12.
- $24 = 12 \times 2$ and $60 = 12 \times 5$.
- So $24 + 60 = 12(2 + 5)$. Check: $12 \times 7 = 84$ and $24 + 60 = 84$.
Answer: $12(2 + 5)$
Worked example: With a letter
Factor $18 + 12x$.
- The GCF of 18 and 12 is 6.
- $18 = 6 \times 3$ and $12x = 6 \times 2x$.
- So $18 + 12x = 6(3 + 2x)$.
Answer: $6(3 + 2x)$
Common mistake
Pulling out a common factor that is not the greatest. $24 + 60 = 2(12 + 30)$ is true, but 12 and 30 still share a factor of 6, so it is not done. Quick check: the two numbers left inside the parentheses should have no common factor other than 1.