How to solve for the sides of a rectangle whose sides are natural numbers given its area is a known natural number?

Question

This may not be the best way to formulate the question but I am looking for a method to solve the following equation $d \cdot n'=n$ where $d, n', n \in \mathbb{N}$ and $n \neq1$ is known. How should I approach this problem? Are there guaranteed solutions?

Answer 1

They would be any two factors of $n$.

• If $n$ is prime, then the numbers are $1,n$.
• If $n=1$, $d=n'=1$.
• If $n$ is composite, take any two factors $d, n'$ of $n$ such that $dn' = n$.

Solutions are indeed guaranteed for all $n \in \mathbb{N}$.

This follows from the facts that by definition the factors of a natural number are in turn also natural numbers, and that all numbers are prime, composite, or $1$.