The table below shows some square numbers and the corresponding numbers on the seven-clock (representing these numbers modulo 7). This works like the days of the week.
| Square numbers in base ten | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| Square numbers modulo 7 | 1 | 4 | 2 | 2 | 4 | 1 | 0 | 1 | 4 | 2 |
For example we say 25 = 4 (mod 7) because when counting up to 25 around the clock you get to the number 4. To avoid lots of counting simply divide 25 by 7 to get 3 remainder 4. Modulus (or clock) arithmetic uses the remainders when one number is divided by another.
Take the number 11 and calculate 1 2, 2 2,
up to 10 2 modulo 11.
Take the number 13 and calculate 1 2, 2 2, up
to 12 2 modulo 13.
What do you notice? What else can you say?