__FACTORS AND MULTIPLES__

A number X that can divide another number Y without a remainder is said to be a factor of Y. e.g. 3 can divide 12 without a remainder, then 3 is a factor of 12. It should be noted that 1 is a factor of every number, a number is also a factor of itself.

**Examples**

List all the factors of the numbers below. (a) 18 (b) 70 (c) 150

__Solution__

** 1. 18**

Follow these steps to list the factors correctly without missing any.

**1 is the number one factor, when used it to divide 18, you will also get 18. That means 1 and 18 are factors. We have **1 and 18**

**Next we try 2, since 2 can divide 18 nine times, that means 2 and 9 are factors also. We have **1, 2, 9, and 18**.

**Next we try 3, it will divide 18 six times, that means 3 and 6 are factors. We will now insert these two numbers into the four numbers we have from the previous step. This will give us **1, 2, 3, 6, 9, and 18**

** Next we try 4, 4 will divide 18 four times (which is 16) remaining 2. Since there is a remainder, it means 4 is not a factor of 18.

** We try 5 also, when we use 5 to divide 18, we have 3 (which is 15) remainder 3. A remainder means 5 is not a factor.

** We now move to 6. A look at our last list of factors shows that 6 is already listed, that means there are no new factors to be added. We will now conclude that the factors of 18 are **1, 2, 3, 6, 9, and 18**

** NOTE: A strong knowledge of the multiplication table is required here to make the listing of factors easier.**

***** TIPS : **

**2 will divide a number if it ends with 2 or zero.****3 will divide a number if the sum of the digits of the number sum up to give a number that is divisible by 3.****4 will divide a number if it gives even number when divided by 2. e.g. 72 when divided by 2 gives 36 (even number ) that means 4 can divide 72 but 70 when divided by 2 gives 35 ( odd number), it means 4 is not a factor of 70****5 will divide a number if it ends with 5 or 0**

** 1. 70**

** 1 and 70 are factors.

** 2 and 35 are factors, our list will be **1, 2, 35, 70**

** 3 is not a factor

** 4 is not a factor

** 5 and 14 are factors, we now have **1, 2, 5, 14, 35, 70**

** 6 is not a factor

** 7 and 10 are factors, we have **1, 2, 5, 7, 10, 14, 35, 70**

** 8 is not a factor

** 9 is not a factor

** 10 is already in our last list, it means there is no new number to be added.

Therefore, factors of 70 are **1, 2, 5, 7, 10, 14, 35 and 70.**

** 3. 150**

Factors are **1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75 and 150.**

** **

From the example above, observe that we listed the factors directly without passing through the long process of example 1 and 2. Once you are perfect with your listing, there is no need to pass through the long process of example 1 and 2.

__Exercise A__

List all the factors of the numbers below:

1. 108

2. 72

3. 360

4. 81

5. 65

6. 31

7. 148

8. 97

9. 117

10. 306

**Prime factors: **After listing all the factors of a given number, the prime number among the listed factors are the prime factors e.g.

** **List the prime factors of 48

__Solution__

First list all the factors of 48

48 – 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Of the listed factors, the prime numbers among them are 2 and 3.

Therefore the prime factors of 48 are 2 and 3.

** Example 2:**

Find the prime factors of 210.

__Solution__

List the factors of 210

1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210

Prime factors are 2, 3, 5 and 7

** Exercise B**

List the prime factors of the following numbers

1. 40

2. 21

3. 55

4. 322

5. 90

6. 124

7. 408

8. 66

9. 207

10. 175

**NEXT:** ** **Common factors, expressing numbers as products of prime factors