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?

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. . . .

A collection of resources to support work on Factors and Multiples at Secondary level.

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. . . .

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?

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

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

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

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?

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

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

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

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

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

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.

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.

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?

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.

Given the products of diagonally opposite cells - can you complete this Sudoku?

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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

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

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

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.

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

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.

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

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

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?

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?

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?

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

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

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?

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

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

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

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

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.

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

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

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. . . .

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

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?

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.

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