A useful shortcut

One way to find a standard name for a number on the clock is to count around the clock. For big numbers this is pretty slow, and it's a little confusing for negative numbers.

Notice that any multiple of 12 is the same as 0: counting 12 takes you exactly once around the circle. So if a number is divisible by 12, its standard name is 0.

For example, 12 = 0;
24 = 0;
1236 = 0;
-12 = 0;
-72 = 0.

Numbers that are not divisible by 12 can be split into two parts: one part that is a multiple of 12, and the remainder. For example,

38 = 3 x 12 + 2 (2 is the standard name for 38) -28 = (-2) x 12 + (-4) (Since -4 = 8, 8 is the standard name for -28.)

The easiest way to find the remainder after taking out multiples of 12 is long division.

38 / 12 = 3 r. 2 -28 / 12 = -2 r. -4

Back to modular arithmetic page.

html">modular arithmetic page.