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

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

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

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

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

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

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

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

Can you work out how many lengths I swim each day?

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

Nine squares are fitted together to form a rectangle. Can you find its dimensions?

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

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

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?

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.

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

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.

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

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

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

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

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

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

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

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?

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

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...

You'll need to know your number properties to win a game of Statement Snap...

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. . . .

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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?

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...

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?

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

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

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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

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

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

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

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?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

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

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

Play this game and see if you can figure out the computer's chosen number.

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.

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!?

Explore the relationship between simple linear functions and their graphs.

Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?