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

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

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

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?

Have you seen this way of doing multiplication ?

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. . . .

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

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.

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

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.

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). . . .

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

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

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

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

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...

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

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.

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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

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.

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

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

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. . . .

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

Explore the relationship between simple linear functions and their graphs.

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?

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?

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

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

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

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

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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

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

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. . . .

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

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

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 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?

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.

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

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

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?