Finding Factors

We can use the prime factorization of a number to find and identify the factors of that number.

Tap on the numbers that you think might be factors of . (Hint: the prime factorization of is ).

Don’t worry about guessing wrong. You will learn how to identify factors next.

Identifying Factors

When guessing the factors of , did you notice a pattern? The prime factorization of can be found inside of the prime factorization of —and is a factor of .

On the other hand, the prime factorization of cannot be found inside of the prime factorization of —and is not a factor of .

Does this pattern hold every time? Can we use it to identify the factors of a number?

Let’s find out by testing if 42 is a factor of 378.

The prime factorization of 378 is 2 × 3 × 3 × 3 × 7 and the prime factorization of 42 is 2 × 3 × 7. Tap on the prime factors of 42 that are in the prime factorization of 378:

Because the prime factorization of 42 is inside of the prime factorization of 378, we can write 378 as the product of 42 and another integer: 378 = 2 × 3 × 3 × 3 × 7 = (2 × 3 × 7) × (3 × 3) = 42 × 9. In this case, 42 is a factor of 378 because it divides evenly into 378 nine times.

We know that number A is a factor of number B if the prime factorization of A is inside of the prime factorization of B because B will be divisible by A. But if the prime factorization of A is not inside of the prime factorization of B, then number A is not a factor of number B. The pattern holds every time.

Practice Identifying Factors

Tap on the factors of ().

Practice Finding Factors

Find eight factors of ().

Use the keypad to enter the prime factorizations of each factor.

Finding All Factors

Find all of the factors of ().

Use the keypad to enter the prime factorizations of each factor.

Calculating the Total Number of Factors

There is a way to calculate the total number of factors from a number’s prime factorization.

The prime factorization of 9450 is 2 × 3 × 3 × 3 × 5 × 5 × 7.

We can find factors of 9450 by combining prime factors—we just have to make sure that our prime factors fit inside of the prime factorization of 9450. That means our factors can’t have more than one 2, three 3’s, two 5’s, or one 7.

For example, if we pick one 2, two 3’s, zero 5’s, and one 7, our factor is 2 × 3 × 3 × 7 = 126. But if we pick one 2, one 3, two 5’s, and one 7, our factor is 2 × 3 × 5 × 5 × 7 = 1050. How many different combinations are there?

One way to generate a random factor of 9450 is by randomly picking the number of 2’s, 3’s, 5’s, and 7’s in its prime factorization.

Number of 2’s:
Number of 3’s:
Number of 5’s:
Number of 7’s:

To see these random numbers more easily, write the factor’s prime factorization using exponents:

2 × 3 × 5 × 7 =

When we pick the number of 2’s in the factor’s prime factorization, we have two choices: 0 or 1. For the number of 3’s, we have four choices: 0, 1, 2, or 3. We have three choices for the number of 5’s (0, 1, or 2) and two choices for the number of 7’s (0 or 1).

To calculate the number of possible combinations, all we have to do is multiply together the number of possible choices: 2 × 4 × 3 × 2 = 48. This means 9450 has 48 factors, which includes 1 (zero 2’s, zero 3’s, zero 5’s, and zero 7’s) and itself.