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 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.

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

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?

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. . . .

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

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?

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

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?

What is the smallest number of answers you need to reveal in order to work out the missing headers?

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.

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?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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...

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

A game that tests your understanding of remainders.

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

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

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

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?

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

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?

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?

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

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

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

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

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?

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

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?

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

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

Are these statements always true, sometimes true or never true?

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

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?

Can you find what the last two digits of the number $4^{1999}$ are?

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

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!?

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

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 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?

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

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

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?

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?