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.

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?

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

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

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

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

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

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

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

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

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.

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

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

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?

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

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

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.

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). . . .

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

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?

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

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

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

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?

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

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...

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

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

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 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?

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

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

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

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Can you make lines of Cuisenaire rods that differ by 1?

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

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

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

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

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

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

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

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

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?

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?

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

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?

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