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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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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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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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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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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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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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Here is a chance to create some Celtic knots and explore the mathematics behind them.

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

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Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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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 produce convincing arguments that a selection of statements about numbers are true?

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

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

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

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

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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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Explore the relationship between simple linear functions and their graphs.

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

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

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

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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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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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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.

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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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

This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.

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Using your knowledge of the properties of numbers, can you fill all the squares on the board?