A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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

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

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 find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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?

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?

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.

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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

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

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?

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

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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

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

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

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

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...

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

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

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.

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

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

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?

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.

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.

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

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.

An environment which simulates working with Cuisenaire rods.

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?"

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

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.

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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.

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

Can you work out how many lengths I swim each day?

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. . . .

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

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

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

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