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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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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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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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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). . . .

Challenge Level

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

Challenge Level

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

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

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

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

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

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

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

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

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

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

Challenge Level

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...

Challenge Level

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

Challenge Level

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

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

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

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

Challenge Level

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

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

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What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

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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.

Challenge Level

Explore the relationship between simple linear functions and their graphs.

Challenge Level

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!?