Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

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

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...

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.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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). . . .

Have you seen this way of doing multiplication ?

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

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

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer 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.

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

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?

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?

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

An introduction to the ideas of public key cryptography using small numbers to explain the process. In practice the numbers used are too large to factorise in a reasonable time.

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. . . .

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

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

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

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.

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

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

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

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

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 noughts are at the end of these giant numbers?

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

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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

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

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.

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?

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.

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.

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

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