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Prime Factorization

Prime Numbers

A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1.
The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17 ..., and we have a prime number chart if you need more.

Factors

"Factors" are the numbers you multiply together to get another number:
factors

Prime Factorization

"Prime Factorization" is finding which prime numbers multiply together to make the original number.
Here are some examples:

Example 1: What are the prime factors of 12 ?

It is best to start working from the smallest prime number, which is 2, so let's check:
12 ÷ 2 = 6
Yes, it divided evenly by 2. We have taken the first step!
But 6 is not a prime number, so we need to go further. Let's try 2 again:
6 ÷ 2 = 3
Yes, that worked also. And 3 is a prime number, so we have the answer:
12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right.

Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 = 22 × 3

Example 2: What is the prime factorization of 147 ?

Can we divide 147 evenly by 2? No, so we should try the next prime number, 3:
147 ÷ 3 = 49
Then we try factoring 49, and find that 7 is the smallest prime number that works:
49 ÷ 7 = 7
And that is as far as we need to go, because all the factors are prime numbers.
147 = 3 × 7 × 7
(or 147 = 3 × 72 using exponents)

Example 3: What is the prime factorization of 17 ?

Hang on ... 17 is a Prime Number.
So that is as far as we can go.
17 = 17

Another Method

We showed you how to do the factorization by starting at the smallest prime and working upwards.
But sometimes it is easier to break a number down into any factors you can ... then work those factor down to primes.

Example: What are the prime factors of 90 ?

Break 90 into 9 × 10
  • The prime factors of 9 are 3 and 3
  • The prime factors of 10 are 2 and 5
So the prime factors of 90 are 3, 3, 2 and 5

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