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

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.

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!?

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?

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?

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

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

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

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

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

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

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

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?

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" .

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.

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

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?

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

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?

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

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. . . .

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

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?

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

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

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

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?

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. . . .

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.

A game that tests your understanding of remainders.

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.

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

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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

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

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

Explore the relationship between simple linear functions and their graphs.

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

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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. . . .

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

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

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?