Calculate gcd(a,b)

Use the following example: $\gcd(2n+3,\,5n-4)=d$.

Since $d\mid 2n+3$, multiply by $5$: $\ d\mid 10n+15$.

Since $d\mid 5n-4$, multiply by $2$: $\ d\mid 10n-8$.

Subtracting gives $d\mid 23$, so $d=1$ or $23$.

At the end, we should find at least one value of $n$ for each possible value of $d$ to prove that the value of $d$ is reachable.

Examples: $\ n=1:\ \gcd(5,1)=1;\qquad n=10:\ \gcd(23,46)=23$.