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.

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

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.

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?

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

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?

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

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

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. . . .

Can you explain the strategy for winning this game with any target?

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

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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?

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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?

Can you find a way to identify times tables after they have been shifted up?

A game that tests your understanding of remainders.

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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

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...

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

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

Given the products of diagonally opposite cells - can you complete this Sudoku?

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. . . .

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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

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

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

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

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Find the frequency distribution for ordinary English, and use it to help you crack the code.

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.

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?

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 are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

Have you seen this way of doing multiplication ?

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

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

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

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

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

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?