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. . . .

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?

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 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.

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?

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

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

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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

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.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

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

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

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...

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.

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

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

Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)

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

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

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

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

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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.

How many zeros are there at the end of the number which is the product of first hundred positive integers?

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

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?

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

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

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

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

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

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Substitution and Transposition all in one! How fiendish can these codes get?

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?

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

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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

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. . . .

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

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

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

Have you seen this way of doing multiplication ?

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