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. . . .

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

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

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

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

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.

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.

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

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?

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?

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

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. . . .

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

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

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.

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

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.

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?

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?

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

Have you seen this way of doing multiplication ?

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

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?

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...

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.

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 sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

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

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

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 game that tests your understanding of remainders.

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.

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. . . .

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

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

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

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

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

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

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?

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

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?

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

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

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

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