Modular arithmetic

problem

Rational Round

Age

16 to 18

Challenge level

3 out of 3

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

Pythagoras Mod 5

Age

16 to 18

Challenge level

3 out of 3

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.
problem

Modular Fractions

Age

16 to 18

Challenge level

3 out of 3

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.
article
Latin Squares

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.
article
An Introduction to Modular Arithmetic

An introduction to the notation and uses of modular arithmetic
article
The Chinese Remainder Theorem

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."
article
Modulus Arithmetic and a Solution to Differences

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

Tom writes about expressing numbers as the sums of three squares.
article
The Knapsack Problem and Public Key Cryptography

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.