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

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

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?

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?

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

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.

Have you seen this way of doing multiplication ?

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

A game in which players take it in turns to choose a number. Can you block your opponent?

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

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!?

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

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

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. . . .

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

A game that tests your understanding of remainders.

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

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.

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?

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?

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 adjacent cells, can you complete this Sudoku?

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

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?

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

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

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

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

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

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.

A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?

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

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

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

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

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

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.

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

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

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

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?

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

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.

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

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

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?