Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

A game in which players take it in turns to choose a number. Can you block your opponent?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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?

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Follow this recipe for sieving numbers and see what interesting patterns emerge.

Is there an efficient way to work out how many factors a large number has?

Can you describe this route to infinity? Where will the arrows take you next?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

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

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can you find the factors?

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

The clues for this Sudoku are the product of the numbers in adjacent squares.

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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?

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

A collection of short problems on factors, multiples and primes.