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

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

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.

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

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.

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.

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?

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

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. . . .

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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?

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?

A game that tests your understanding of remainders.

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.

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?"

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

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

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

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. . . .

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

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

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?

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...

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Given the products of adjacent cells, can you complete this Sudoku?

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

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

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?

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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

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

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

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.

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. . . .

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.

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. . . .

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

Find the frequency distribution for ordinary English, and use it to help you crack the code.

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

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

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.

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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

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

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

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