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# Prime Factorization – Explanation & Examples

**Prime factorization** is a method of finding all the prime numbers that multiply to form a number. Factors are multiplied to get a number, while prime factors are the numbers that can only be divided by 1 or themselves.

## How to find Prime Factorization?

There are two methods of finding prime factors of a number. These are repeated division and factor tree.

### Repeated division

A number is reduced by dividing it severally with prime numbers. Prime factors of number 36 are found by repeated division as shown:

The prime factors of number 36 are, therefore, 2 and 3. This can be written as 2 × 2 × 3 × 3. It is advisable to start dividing a number by the smallest prime number and proceed to bigger factors.

*Example 1*

*What are the prime factors of 16?*

__Solution__

The best way to solve this problem is by identifying the smallest prime factor of the number, which is 2.

Divide number by 16;

16 ÷ 2 = 8

Because 8 is not a prime number, proceed by dividing again by the smallest factor;

8 ÷ 2 = 4

4 ÷ 2 = 2

We have the prime factors of 16 highlighted in yellow, and they include: 2 x 2 x 2 x 2.

which can be written as an exponent:

16 = __2 ^{2}__

* *

*Example 2*

*Find the prime factors of 12.*

__Solution__

Divide 12 by 2;

12 ÷ 2 = 6

6 is not prime, proceed;

6 ÷ 2 = 3.

Therefore, 12 = 2 x 2 x 3

12 = 2 ^{2 }× 3

It is noted that, all prime factors of a number are prime.

*Example 3*

*Factorize 147.*

__Solution__

Start by dividing 147 by the smallest prime number.

147 ÷ 2 = 73.5

Our answer isn’t an integer, try the next prime number 3.

147 ÷ 3 = 49

Yes, 3 worked, now proceed to the next prime that can divide 49.

49 ÷ 7 = 7

Therefore, 147 = __3 x 7 x 7,__

=__3 x 7 ^{2}.__

* *

*Example 4*

*What is the prime factorization of 19?*

19 =__ 19__

__Solution__

Another method on how to perform factorization is to break a number down into two integers. Now find the prime factors of the integers. This technique is useful when dealing with bigger numbers.

* *

*Example 5*

*Find the prime factors of 210.*

__Solution__

Break down 210 into:

210 = 21 x 10

Now calculate the factors of 21 and 10

21 ÷ 3 = 7

10 ÷ 2 = 5

Combine the factors: 210 = __2 x 3 x 5 x 7__

### Factor tree

Factor tree involves finding the prime factors of a number by drawing tree- like programs. Factor tree is the best tool of doing prime factorization. The prime factors of 36 are obtained by factor tree as shown below:

## Practice problems

1. *The following are the prime factorization of certain numbers. Calculate the number.*

(i) 3× 5 × 11

(ii) 2 × 5 × 7

(iii) 2 × 3 × 13

(iv) 2 × 3 × 3 × 7

(v) 3 × 7 × 11

(vi) 3 × 5 × 5

(vii) 2 × 3 × 7

(viii) 2 × 2 × 3 × 11

(ix) 3 × 7 ×11 × 11

2. *Determine the prime of these numbers by division method.*

(i) 56

(ii) 38

(iii) 12

(iv) 120

(v) 64

(vi) 49

(vii) 81

(viii) 21

3. **Using factor method, determine the prime factors of:**

(i) 70

(ii) 11

(iii) 99

(iv) 44

(v) 62

(vi) 76

(vii) 97

(viii) 63

4. **Factorize by any method.**

(i) 9

(ii) 63

(iii) 90

(iv) 48

(v) 34

(vi) 40

(vii) 66

(viii) 88

(ix) 52

(x) 98

(xi) 75

(xii) 100

5. **What are the prime factors of 19?**

a. 19

b. 0

c. 2 x 9.5

d. None of the above

6. *What are the prime factors of 50?*

a. 2 x 2 x 12.5

b. 2 x 25

c. 2 x 5x 5

d. 1 x 2 x 5 x 5

7.** Calculate the prime factors of 25.**

a. 2 x12.5

b. 5 x 5

c. 1 x 25

d. 5 x 5.5

**8**. **Find the prime factors of 81.**

a. 3 x 2 7

b. 3 x 3 x 3 x3

c. 9 x 9

d. None of the above

9. *Determine all the prime factors of 125.*

a. 1 x 125

b. 5 x 5 x 5

c. 2 x 5 x 12.5

d. All of the above

10. *Calculate the prime factors of 132.*

a. 2 x 2 x 3 x 11

b. 2 x 6 x 11

c.2 x 2 x 2 x 3 x 11

d. 4 x 3 x11

__Answers__

- (i)
__165__

(ii)__ 70__

(iii) __78__

(iv) __126__

(v) __231__

(vi) __75__

(vii) __42__

(viii)__ 132__

(ix) __2541__

- (i)
__2__^{2 }× 7

(ii) __2 × 19__

(iii) __2 × 2 x 3__

(iv) __2 ^{3 }x 3 x 5__

(v) __2 ^{6}__

(vi)__ 7 x 7__

(vii) __3 x 3 x 3 x 3__

(viii)__ 3 × 7__

- (i)
__2 × 5 x 7__

(ii) __11__

(iii)__ 3 x 3 x 11__

(iv) __2 x 2 x 11__

(v)__ 2 × 31__

(vi) __2 × 2 × 19__

(vii) __97__

(viii) __3 x 3 x 7__

- (i)
__3 x 3__

(ii) __3 x 3 x 7__

(iii) __2 x 3 x 3 x 5__

(iv) __2 × 2 x 2 x 2 x 3__

(v) __2 × 17__

(vi) __2 × 2 × 2 x 5__

(vii) __2 × 3 × 11__

(viii) __2 × 2 × 2 × 11__

(ix) __2 x 2 x 13__

(x) __2 × 7 x 7__

(xi) __3 x 5 x 5__

(xii) __2 x 2 x 5 x 5__

- Answer
__19__ - Answer
__2 x 5 x 5__ - Ans.
__5 x 5__ - Ans.
__3 x 3 x 3 x 3__ - Ans.
__5 x 5 x 5__ - Ans
__. 2 x 2 x 3 x 11__

- Answer