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

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. . . .

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?

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

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

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

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. . . .

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?

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

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?

Have you seen this way of doing multiplication ?

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

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.

A collection of resources to support work on Factors and Multiples at Secondary level.

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

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?

Can you find what the last two digits of the number $4^{1999}$ are?

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.

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

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?

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). . . .

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

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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

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

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

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

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

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?

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

Explore the relationship between simple linear functions and their graphs.

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. . . .

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

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. . . .

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

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.

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?

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

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

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

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

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.

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.

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?

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

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