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?

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.

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

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

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

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?

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?

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?

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

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

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

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?

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

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

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?

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

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

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

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

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

Number problems at primary level that may require resilience.

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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

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. . . .

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

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

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

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

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.

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

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.

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

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?

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

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

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

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

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?

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

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.

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

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

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...

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

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

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

Substitution and Transposition all in one! How fiendish can these codes get?

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.