Modular arithmetic

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

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

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

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?

Here are many ideas for you to investigate - all linked with the number 2000.

Add powers of 3 and powers of 7 and get multiples of 11.

What is the smallest perfect square that ends with the four digits 9009?

What is the remainder when 2^{164}is divided by 7?

What is the units digit for the number 123^(456) ?

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.