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## 'Clock Squares' printed from http://nrich.maths.org/

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?