A Survey of Mathematics Related Topics: Pg. 496, Clock Arithmetic

Clock time can be thought of as a type of modular arithmetic. Modular arithmetic is a method of finding remainders. On most standard clock faces there are 12 hours displayed. If it is three o'clock and you want to know what time it will be 10 hours from now you would add 10 hours to three o'clock. You can do this by moving the hour hand of the clock around the clock face 10 hours. Notice when you do this the clock says it is 1 o'clock.

But 3 + 10 = 13 ?

True 3 + 10 = 13 but when using modular arithmetic or in this case "clock arithmetic" your final number can never be larger than the largest number displayed on the clock (N), so the remainders wrap around the clock..

In this example the largest number was 12 so the remaining number wrapped around and became 1.

Here are two rules to keep in mind.

1. When we add or multiply two numbers and the result is less than N, the final answer is just the number resulting from the addition or multiplication.

2. When we add or multiply two numbers and the result is more than N, the final answer is found by thinking of the numbers as looping back to the beginning, i.e., N = 0, N + 1 = 1, N + 2 = 2, etc.

A clock with 5 hours works the same way: 17 hours after 0 o'clock is 2 o'clock (17 = 5 + 12 = 10 + 7 = 15 + 2 which is 2 on the dial).