Review question

# Can we express this unit fraction as the sum of two others? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9583

## Question

A number of the form $1/N$, where $N$ is an integer greater than $1$, is called a unit fraction.

Noting that $\frac{1}{2} = \frac{1}{3} + \frac{1}{6}\quad\mbox{and}\quad\frac{1}{3}=\frac{1}{4}+\frac{1}{12},$ guess a general result of the form $\frac{1}{N} = \frac{1}{a}+\frac{1}{b}$ and hence prove that any unit fraction can be expressed as the sum of two distinct unit fractions.

By writing the previous equation in the form $(a − N)(b − N) = N^2$ and by considering the factors of $N^2$, show that if $N$ is prime, then there is only one way of expressing $1/N$ as the sum of two distinct unit fractions.

Prove similarly that any fraction of the form $2/N$, where $N$ is prime number greater than $2$, can be expressed uniquely as the sum of two distinct unit fractions.