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?