Take, for example, the two numbers 4 and 6. Both numbers are divisible by 2, so 2 is a common factor of 4 and 6.
As you get more experience working with numbers, you will often be able to find common factors by inspection. This means you can look at the two numbers and figure out quickly what is the common factor.
If you don't have enough experience yet to find common factors by inspection, you can use prime factorization to find common factors. The method most often used to get the prime factorization of a number is a factor tree. Figure 1 is an example of a factor tree.
What are the common factors of 30 and 45?
Step 1: Factor 30 and 45.
|Figure 2: Factor tree of 30||Figure 3: Factor tree of 45|
Step 2: Find the common prime factors.
|Figure 4: Factor tree of 30.||Figure 5: Factor tree of 45.|
Step 3: Calculate all common factors.
The common factors are 3, 5, 3 · 5 = 15.
An expression can also be a common factor. For example, the expression x - 2 is a factor of the expression (x - 2)(x + 1). Since (x - 2)(x + 1) = x2 - x - 2, the expression x - 2 is also a factor of the polynomial x2 - x - 2.
What are the common factors, if any, of (x - 2)(x + 1)(x + 4) and (x - 4)(x + 1)(x + 4)? Since (x + 1) and (x + 4) are factors of the two expressions and are common (meaning the same), (x + 1) and (x + 4) are common factors of (x - 2)(x + 1)(x + 4) and (x - 4)(x + 1)(x + 4).
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