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Glossary

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Divisibility and PrimesFactors and Multiples

By now you should be comfortable with addition, subtraction and multiplication of integers. Division is slightly different, because you can’t always divide any integer by any other. For example 17 divided by 3 is not a whole number – it is somewhere in between 5 and 6. You either have to give a remainder (2), or express the answer as a decimal number (5.66).

0 1 2 3 4 5 6 7 8 9 10 11 12 3 3 12 3 3

12 is divisible by 3

0 1 2 3 4 5 6 7 8 9 10 11 12 4 4 4 10

10 is not divisible by 4

If you can divide a number A by a number B, without remainder, we say that B is a factor (or divisor) of A, and that A is a multiple of B. We often write B|A, where the vertical bar simply means “divides”.

For example, 7 × 3 = 21, so 7 is a of 21, 21 is a of 7, and 7|21.

In this short game you have to determine which numbers are factors or multiples, as fast as possible. Click the play button to start.

Factors and Multiples Quiz

${x}
is a
factor
multiple
neither
of
${y}

It is often useful to find all the divisors of a number. For example, the divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Of course, you don’t want to check all numbers up to 60 if they are divisors. Instead, there is a simple technique which relies on the fact that divisors always appear in .

In the case of 60 we have 60 = 1 × 60 = 2 × 30 = 3 × 20 = 4 × 15 = 5 × 12 = 6 × 10. Or, in a different notation,

601,2,3,4,5,6,10,12,15,20,30,60

To find all divisors of a number we simply start at both ends of this list, until we meet in the middle.

421,2,3,6,7,14,21,42
For example, the first divisor pair of 42 is simply 1 and 42, and we write them down with much space in between.
After 1 at the beginning, we check if 2 divides 42. It does, and the corresponding pair is 42 ÷ 2 = 21.
Next, we check if 3 divides 42. It does, and the corresponding pair if 42 ÷ 3 = 14.
Now we check if 4 divides 42. It does not, however, so we move on.
5 also doesn’t divide 42 so we move on.
6 does divide 42 again. Its pair is 42 ÷ 6 = 7. Notice how we’ve met in the middle after only a few attempts, without having to test all numbers from 7 to 42.

The only special case with this method is for square number: in that case, you will meet at just a single number in the middle, like 64 = 8 × 8.