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

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

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

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

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

Which numbers can we write as a sum of square numbers?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

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

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

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

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

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

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine. . . .

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.

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check. . . .

In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

How many different ways can you arrange the officers in a square?

What remainders do you get when square numbers are divided by 4?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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?

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: x2 and -5. What do you think?