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

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?

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?

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

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

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

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

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

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?

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

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

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!?

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

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

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

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

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

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

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.

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

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn.

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?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

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

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

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?

Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

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?

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

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...

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

This article for teachers describes how number arrays can be a useful reprentation for many number concepts.

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

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

I am thinking of three sets of numbers less than 101. They are the red set, the green set and the blue set. Can you find all the numbers in the sets from these clues?

I am thinking of three sets of numbers less than 101. Can you find all the numbers in each set from these clues?

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

What is the smallest number of answers you need to reveal in order to work out the missing headers?

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

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

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?

Are these statements always true, sometimes true or never true?

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. . . .

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?