relation between exponent values and solution 
If $a,b,c$ and $d$ are positive integers such that $a^5=b^6$ and $c^3=d^4$ and $d-a=61$, then find the smallest value for $c-b$.

$d-a=61$, so $d=a+61$ or $a=61-d$.  But then how to substitute these values in the equations and solve? I am stuck.

Notice that $d$ must be a cube greater than 61, so let $d=n^3$, where $n \geq 4$; hence $c=n^4$ and $a=n^3-61$.
Also $a=b$. Thus you need to minimize $n^4-n^3+61$ subject to $n \geq 4$, which happens at $n=4$.
