Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check. . . .

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

Which numbers can we write as a sum of square numbers?

A weekly challenge concerning the interpretation of an algorithm to determine the day on which you were born.

Choose any whole number n, cube it, add 11n, and divide by 6. What do you notice?

What day of the week were you born on? Do you know? Here's a way to find out.

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.

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine. . . .

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.

What are the possible remainders when the 100-th power of an integer is divided by 125?

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

What is the smallest perfect square that ends with the four digits 9009?

Learn about the rules for a group and the different groups of 4 elements by doing some simple puzzles.

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

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?

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

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

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?

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

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

How many different ways can you arrange the officers in a square?

What can you say about the common difference of an AP where every term is prime?

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

What remainders do you get when square numbers are divided by 4?

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

An example of a simple Public Key code, called the Knapsack Code is described in this article, alongside some information on its origins. A knowledge of modular arithmetic is useful.

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.