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

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.

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" .

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

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

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?

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.

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 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...

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?"

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?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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

Can you explain the strategy for winning this game with any target?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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.

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

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

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

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

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

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

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

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. . . .

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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?

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

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

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. . . .

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

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

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!?

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

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?

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?

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.

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?

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

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.

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

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?

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

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

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