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

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?

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

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.

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

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

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

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?

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?

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!?

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

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.

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. . . .

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

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 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?

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

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

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

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?

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

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

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

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

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

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

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?

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

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

A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.

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. . . .

A game that tests your understanding of remainders.

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?

Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

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

These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

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?

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?

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

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

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

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

Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off?

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

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?"

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