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 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?

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...

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?

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?

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

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?

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. . . .

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

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

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

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

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.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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?

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

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?

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

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?

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

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

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

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

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

A game that tests your understanding of remainders.

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

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.

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?

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

Find the frequency distribution for ordinary English, and use it to help you crack the code.

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

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?

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.

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

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?

Substitution and Transposition all in one! How fiendish can these codes get?

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?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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

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

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