# Prime Factorization

This is a complete lesson with instruction and exercises about prime factorization, meant for 4th or 5th grade. It first briefly reviews what are primes, and then explains how to factor numbers using a factor tree. After several examples, there are many factorization exercises for the students.

Then the lesson explains how all numbers are "built" from primes, and includes exercises about that process.

 Some numbers only have two divisors: 1 and the number itself. Such numbers are called prime numbers. 11 is one of them. factor factor product 1 × 11 = 11

In the last lesson, we found that the prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. One is usually not counted as a prime number.

## Prime factorization using a factor tree

A factor tree is a handy way to factor numbers to their prime factors. The factor tree starts at the root and grows upside down!

We want to factor 24 so we write 24 on top. First, 24 is factored into 4 × 6. However, 4 and 6 are not primes, so we can continue factoring. Four is factored into 2 × 2 and six is factored into 2 × 3.

We will not factor 2 or 3 any further because they are prime numbers.

 (root) 24 /     \ 4 × 6 /  \ /  \ (leaves) 2 × 2 × 2 × 3

Once you get to the primes in your "tree", they are the "leaves", and you stop factoring in that “branch”. So 24 = 2 × 2 × 2 × 3. This is the prime factorization of 24.

Examples:

 30 /   \ 5 × 6 /   \ 2 × 3
5 is a prime number—it is a "leaf". Once done, "pick the leaves"—you can even circle them to see them better! So, 30 = 2 × 3 × 5.
 21 /  \ 3 × 7
Both 3 and 7 are prime numbers, so we cannot factor them any further.
So 21 = 3 × 7.
 66 /   \ 11 × 6 /   \ 2 ×  3

OR

 66 /   \ 2 × 33 /   \ 11 × 3
You can start the factoring process any way you want. The end result is the same: 66 = 2 ×  3 × 11.
 72 /      \ 12   ×  6 /    \     /   \ 3 × 4 × 2 × 3 /  \ 2 × 2

72 has lots of factors so the factoring takes many steps.

72 = 2 × 2 × 2 × 3 × 3

We could have also started by writing 72 = 2 × 36 or 72 = 4 ×18.

 57 /   \
How can you get started?
Check:
- is 57 in any of the
times tables?
- is it divisible by 2?
By 3? By 5?

 65 /   \
How can you get started?
Check:
- is 65 in any of the
times tables?
- is it divisible by 2?
By 3? By 5?

1. Factor the following numbers to their prime factors.

 a.  18      / \ b.  6     / \ c.  14      / \ d.  8      / \ e.  12      / \ f.  20      / \ g.  16      / \ h.  24      / \ i.  27      / \ j.   25      / \ k.  33      / \ l.  15      / \

2. Factor the following numbers to their prime factors.

 a.  42      / \ b.  56     / \ c.  68      / \ d.  75      / \ e.  47      / \ f.  99      / \ g.  72      / \ h.  80      / \ i.  97      / \ j.   85      / \ k.  66      / \ l.  82      / \

Prime numbers are like building blocks of all numbers. They are the first and foremost, and other numbers are "built" from them. "Building numbers" is like factoring backwards. We start with the building blocks—the primes—and see what number we get:

 2 × 5 × 2 × 2 \ /         \ / 10 × 4 \         / 40

 2 × 3 × 2 × 3 × 2 \ / \ / | 6 × 6 × 2 | \      / 6 × 12 \             / 72

 5 × 2 × 7 \  /        | 10 × 7 \       / 70

 2 × 7 × 2 × 3 \ / \   / 14 × 6 \        / 84

By using the process above (building numbers starting from primes) you can build ANY whole number there is! Can you believe that?

We can say this in another way: ALL numbers can be factored so the factors are prime numbers. That is sort of amazing! This fact is known as the fundamental theorem of arithmetic. Indeed, it is fundamental.

 992 /      \ 4     ×   248 /   \         /     \ 2  × 2 ×  4 × 62 /   \      /   \ 2 × 2 × 2 × 31

So, no matter what the number is—992 or 83,283 or 150,282—it can be written as a product of primes.

See 992 factored on the right. 992 = 2 × 2 × 2 × 2 × 2 × 31. For 83,283 we get 3 × 17 × 23 × 71, and 151,282 = 2 × 3 × 3 × 3 × 11 × 11 × 23.

To find these factorizations, you need to test-divide the numbers by various primes so it is a bit tedious. Of course, computers can do the divisions very quickly.

3. Build numbers from primes.

 a.  2 × 5  × 11     \    /        | b.  3 × 2 × 2 × 2      \   /       \   / c.  2 × 3  × 7      \   /        | d.  11 × 3 × 2       |       \   / e.  3 × 3 × 2 × 5       \   /       \   / f.  2 × 3 × 17

4. Build more numbers from primes.

 a.   2 × 5 × 13 b.  7 × 13 × 2 × 11 c.  19 × 3 × 5 × 2

5. Try it on your own! Pick 3-6 primes as you wish (you can use the same prime
several times), and see what number is built from them. Ready for a challenge? Use your knowledge of divisibility tests and the calculator, and find the prime factorization of these numbers: a.  2,145 b. 3,680 c. 10,164

This lesson is taken from Maria Miller's book Math Mammoth Multiplication & Division 3, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

#### Math Mammoth Multiplication & Division 3

A self-teaching worktext for 5th grade that covers multi-digit multiplication, long division, problem solving, simple equations, ratios, divisibility, and factoring.