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

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

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

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

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

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6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

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

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

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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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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?

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

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

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

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

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What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

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How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?

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

This article for teachers describes how number arrays can be a useful representation for many number concepts.

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

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What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

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Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

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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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Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

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Norrie sees two lights flash at the same time, then one of them flashes every 4th second, and the other flashes every 5th second. How many times do they flash together during a whole minute?

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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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This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

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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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Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

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There are eight clues to help you find the mystery number on the grid. Four of them are helpful but the other four aren't! Can you sort out the clues and find the number?

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

This article for primary teachers outlines why developing an intuitive 'feel' for numbers matters, and how our activities focusing on factors and multiples can help.

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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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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.