Modular arithmetic

A country has decided to have just two different coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: ×2 and -5. What do you think?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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?

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?