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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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A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

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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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Given the products of diagonally opposite cells - can you complete this Sudoku?

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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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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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In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

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

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

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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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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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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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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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The items in the shopping basket add and multiply to give the same amount. What could their prices be?

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

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

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

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

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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

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

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

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

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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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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 numbers from a sequence. What numbers can we make now?

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

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

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

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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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An environment which simulates working with Cuisenaire rods.

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

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

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

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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. . . .