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

Can you find any perfect numbers? Read this article to find out more...

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

Nine squares are fitted together to form a rectangle. Can you find its dimensions?

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

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

Can you work out what size grid you need to read our secret message?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

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

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Can you find any two-digit numbers that satisfy all of these statements?

Can you work out how many lengths I swim each day?

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

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

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...

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

Play this game and see if you can figure out the computer's chosen number.

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

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

Find the highest power of 11 that will divide into 1000! exactly.

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

You'll need to know your number properties to win a game of Statement Snap...

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

Given the products of adjacent cells, can you complete this Sudoku?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Find the number which has 8 divisors, such that the product of the divisors is 331776.

56 406 is the product of two consecutive numbers. What are these two numbers?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

What is the smallest number of answers you need to reveal in order to work out the missing headers?