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.

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?

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

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

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. . . .

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

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

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

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

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?

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 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?

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?

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

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

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

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

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?

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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.

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

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

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?

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.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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?

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?

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?

A game that tests your understanding of remainders.

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.

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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?

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

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.

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?

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

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?"

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