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

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

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

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!?

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?

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

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?

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

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

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

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

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.

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

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

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

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

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.

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

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

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.

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

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...

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

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

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

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

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

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

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

Can you work out what size grid you need to read our secret message?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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

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

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

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.

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

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

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?

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.

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?

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. . . .

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

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

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

Can you find what the last two digits of the number $4^{1999}$ are?

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?

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.

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.