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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.

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

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

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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.

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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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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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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.

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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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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. . . .

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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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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?

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

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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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Got It game for an adult and child. How can you play so that you know you will always win?

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

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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

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

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

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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...

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

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The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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

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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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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.

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Explore the relationship between simple linear functions and their graphs.

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

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

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

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

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

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

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

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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.

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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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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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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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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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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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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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When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .