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...

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?

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 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). . . .

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?

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

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.

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. . . .

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

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.

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?

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

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.

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

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

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

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

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

Explore the relationship between simple linear functions and their graphs.

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?

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

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

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. . . .

Have you seen this way of doing multiplication ?

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

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. . . .

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.

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?

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

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.

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

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

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.

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

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?

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?

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

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.

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?