Modular arithmetic

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

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.

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?