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

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