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

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.

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?

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

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?

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.

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 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?

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

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

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?

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

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. . . .

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

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

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" .

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

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

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

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

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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

Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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

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.

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. . . .

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

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?

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

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. . . .

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

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

Explore the relationship between simple linear functions and their graphs.

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?

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

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

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

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

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

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

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?"

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