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?

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

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.

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

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

Got It game for an adult and child. How can you play so that you know you will always win?

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?

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

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.

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

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?

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

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

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 added together some of my neighbours house numbers. Can you explain the patterns I noticed?

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

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

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

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

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.

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.

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. . . .

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

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

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?

How many noughts are at the end of these giant numbers?

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

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

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

Can you find any two-digit numbers that satisfy all of these statements?

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

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.

Play this game and see if you can figure out the computer's chosen number.

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

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?"

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

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?

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?

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

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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

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

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

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

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

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

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.