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

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

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?

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 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.

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. . . .

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?

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

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.

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

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 numbers from a sequence. What numbers can we make now?

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

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.

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

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?

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.

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

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?

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

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

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

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...

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

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

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

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

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?

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

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

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. . . .

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...

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.

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

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

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

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

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

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. . . .

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?

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

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

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

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?

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

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?

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