Challenge Level

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.

Challenge Level

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

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

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

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

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

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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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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 zeros are there at the end of the number which is the product of first hundred positive integers?

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

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

Challenge Level

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

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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...

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

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

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

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

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

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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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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?"