Factors and Multiples



Factors

A factor of a number will divide that number exactly, without any remainder.

Example : 2 is a factor of 16 because 2 goes into 16 exactly 8 times, with no remainder.

But note that 16 also has other factors apart from 2. These other factors are

  • 4 : which goes into 16 exactly 4 times
  • 8 : which goes into 16 exactly 2 times
  • as well as 16 itself : 16 divided by 16 equals 1
  • and also 1 : 16 divided by 1 equals 16

So the number 16 has a total of 5 factors.

Example : 24 has the following factors

1,2,3,4,6,8,12,24

Example : 3 is a factor of 9 because 3 divides into 9 exactly, without any remainder.

Example : 3 is also a factor of

27, 18, 30, 15,....and many more

Practice Question

Which of the following are factors of 72 ?

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14

Practice Question

4 is a factor of which of these numbers ?

1, 2, 3, 4, 6, 7, 8, 10, 14, 22, 34, 72, 114, 1064

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Highest Common Factor

The Highest Common Factor (HCF) of two numbers is the highest number that is a factor of both numbers

Example : The numbers 36 and 24 have a common factor of 4. But this not the Highest Common Factor, because 6 is also a common factor. The number 6 itself is still not the required HCF - the HCF is actually 12.

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Finding Factors of Large Numbers

Some rules

First the easier ones

  • 2 is a factor of all even numbers
  • 10 is a factor of all numbers ending in 0
  • 5 is a factor of all numbers ending in a 0 or 5
And a few "not so obvious" rules
  • 3 is a factor of a number if it is also a factor of the sum of the individual digits in the number, e.g
    • 65421 (digits add up to 18)
    • 24 (digits add up to 6)
    • 321 (digits add up to 6)
  • 9 is a factor of a number if it is also a factor of the sum of the individual digits in the number, e.g.
    • 76437 (digits add up to 27)
    • 27 (digits add up to 9)
    • 981 (digits add up to 18)
  • 11 is a factor of a three figure number if the middle digit is the sum of the two outside digits, e.g. 792

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Multiples

Multiples of a number are obtained just by multiplying that number by a whole number.

Example The multiples of 6 are

6, 12 , 18, 24, 30, etc

Note that 6 is a multiple of 6 because it is equal to 6 multiplied by one.

Practice Question

Which of the following are multiples of 7 ?

1, 2, 3, 4, 7, 8, 9, 14, 161, 245, 344, 567, 2996

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Lowest Common Multiple

The Lowest Common Multiple (LCM) of two numbers is a multiple of both numbers and lower than all other numbers that are also multiples of both the numbers.

Example The Lowest Common Multiple of 6 and 4 is 12.

12 is lower than other common multiples like 24, 36, 48, and there is no common multiple of 6 and 4 which is lower than 12.

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