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

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

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, 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?

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

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

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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...

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 can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

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

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

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" .

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?

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

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.

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

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

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

How many noughts are at the end of these giant numbers?

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

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

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.

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.

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

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

Got It game for an adult and child. How can you play so that you know you will always win?

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

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

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

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

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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. . . .

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

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

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. . . .

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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?

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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

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