Modular arithmetic

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

What day of the week were you born on? Do you know? Here's a way to find out.

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Choose any whole number n, cube it, add 11n, and divide by 6. What do you notice?

Can you interpret this algorithm to determine the day on which you were born?