# 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…).

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 **21**. Similarly, **21** is a **7**, and we can write **7**|**21**.

In this short game you have to determine which numbers are factors or multiples:

### Factors and Multiples Quiz

It is often useful to find *all* the factors of a number. For example, the factors 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 factors. Instead, there is a simple technique which relies on the fact that factors 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,

60 | 1, | 2, | 3, | 4, | 5, | 6, | 10, | 12, | 15, | 20, | 30, | 60 |

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

42 | 1, | 2, | 3, | 6, | 7, | 14, | 21, | 42 |

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