You do modular arithmetic several times every day when you are thinking about time. Imagine, for example, you're going on a train trip at 11pm that lasts three hours. What time will you arrive? Not at 11+3 = 14 o'clock, but at 2 o'clock in the morning. That's because, on a 12-hour clock, you start counting from the beginning again after you get to 12. (On a 24-hour clock, you start again after you get to 24.) So, on a 12-hour clock you have:

4 + 9 = 1,

7 + 7 = 2,

5 + 12 = 5,

and so on. When you are subtracting hours, you do the same but backwards:

4 - 7 = 9

1 - 11 = 2

6 - 12 = 6.

You could play the same game using other numbers, apart from 12 and 24, to define your cycle. For example, in modular arithmetic *modulo* 5 you have

4 + 2 = 1

3 + 4 = 2

1 - 4 = 2

3 - 5 = 3

These sums can be a little tedious to work out if you're counting on your fingers, but luckily there is a general method. Let's say you're doing arithmetic modulo some natural number *p* and you're looking at some other natural number *x*. To find the value of *x* modulo *p* (the value of *x* on a clock with *p* hours), compute the remainder when dividing *x* by *p*: that's your result.

This also works when *x* is negative (noting that the remainder is defined to be always positive). For example, for *p* = 12 and *x* = -3 we have

-3 = (-1) x 12 + 9,

so the remainder is 9. Therefore -3 modulo 12 is equal to 9. (If you use the modulus function in some computer languages you have to be a little careful though, as some return a different value for negative numbers.)

If you want to add or subtract two numbers module some natural number *p*, you simply work out the result, call it *x*, in ordinary arithmetic and then find the value of *x* modulo *p*.

There is clearly something very cyclical about modular arithmetic. Whatever number *p* defines your arithmetic, you can think of it as counting forward or backward in clock with *p* hours. To put this in technical maths language, modular arithmetic modulo *p* gives you a cyclic group of order *p*. You can find out more about group theory in this article and about modular arithmetic on our sister site NRICH.

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## Translation

The article translated into Arabic: https://goo.gl/evtTq8