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

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?

• You have basically asked if it is possible to find a factorization of $n$. If you forbid the trivial factorization of $1\cdot n = n$, then this problem is at least as hard as determining if $n$ is prime, and probably not harder than breaking the (multiprime) RSA encryption scheme. – InequalitiesEverywhere Jan 1 '19 at 12:48

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 $$1$$.