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

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?

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

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

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

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.

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.

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

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

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!?

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

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?

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

Can you make lines of Cuisenaire rods that differ by 1?

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. . . .

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...

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

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

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

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

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

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

Follow this recipe for sieving numbers and see what interesting patterns emerge.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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.

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

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

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

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

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

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?

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

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

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.

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

How many noughts are at the end of these giant numbers?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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.

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

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

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?

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...

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...

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?

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?

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