Whole Numbers II

Mathematics J.S.S 2 First Term

Theme: Numbers & Numeration

Sub Theme: Whole Number

WEEK 2

Whole Numbers II

Performance Objectives

Students should be able to;

1. Find prime factors of numbers not greater than 200
2. Express Numbers as product of its prime factors

Prime Factors of Numbers

Content

A prime number is a number divisible only by 1 and itself. Examples are 1, 2, 3, 5…e.t.c

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

       

 

Prime factors of a number are the factors of that number which are not divisible by any other number except that number and 1. For example, the prime factors of 30 are 2,3 and 5 because out of all the factors of 30 (1,2,3,5,6,10,15 and 30), it is only 2,3 and 5 that are not divisible by any other number except themselves and 1.

Examples:

 

Write the factors of the following numbers and state the prime factors, hence express each number as a product of its prime factors.

(a) 22 (b) 50

Solutions:                       

(a) 22

Factors of 22 are 1, 2, 11, 22

The prime factors of 22 are 2 and 11

Product of its prime factors of 22 = 2 × 11

(b) Factors of 50 are 1, 2, 5, 10, 25 and 50

The prime factors of 50 are 2 and 5

Product of its prime factors of 50 = 2 × 5 × 5 = 2 × 5² 

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?

147 ÷ 2 = 73½

No it can’t. The answer should be a whole number, and 73½ is not.

Let’s try the next prime number, 3:

147 ÷ 3 = 49

That worked, now 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

It is by breaking down into any factors and then those factors down to prime.

Examples

Express 27 as a product of their prime factors in index form. 

Solutions:                       

Divide each of the numbers by the prime factors in turns until it will not divide any further.

1. 27

3	27
3	9
3	3
	1

∴ 27 = 3 x 3 x 3 = 3³