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

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. . . .

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

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

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?

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?

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?

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

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

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.

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. . . .

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?

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.

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

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

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

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?

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

For this challenge, you'll need to play Got It! 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?

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.

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

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.

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

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?

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. . . .

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

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?

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

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

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.

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

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

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

A game that tests your understanding of remainders.

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

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

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...

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

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?

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

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). . . .

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

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

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?

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

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