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?

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

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

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

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

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

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. . . .

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?

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

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. . . .

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

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...

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.

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

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?

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

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

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. . . .

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.

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...

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

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?

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

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.

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

Find the number which has 8 divisors, such that the product of the divisors is 331776.

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

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

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

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

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

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

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

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?

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

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

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

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

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

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.

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

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

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

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

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