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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.

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

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

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

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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?

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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?

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

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

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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. . . .

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

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

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

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

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

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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.

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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.

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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). . . .

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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?

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

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

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

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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...

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

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

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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...

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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?

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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?

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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?

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

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

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

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

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

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

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

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

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Explore the relationship between simple linear functions and their graphs.

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

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

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

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

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

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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.

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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.

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

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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.