2006 SMT/Algebra Problems/Problem 7

Problem

Find all solutions to $aabb=n^4-6n^3$, where $a$ and $b$ are nonzero digits, and $n$ is an integer. ($a$ and $b$ are not necessarily distinct.)

Solution

Notice that $aabb=(1100)a+(11)b=(11)(100a+b)$. Therefore, $n^4-6n^3=(n^3)(n-6)$ must be divisible by $11$. Since $11$ is prime, either $n$ or $n-6$ must be a multiple of $11$. However, if $n=17$, then $n^4-6n^3>10000$, so we only need to consider when $n=11$. In this case, $n^4-6n^3=6655$, a solution, so the only solution is $(a,b,n)=\boxed{(6,5,11)}$.

See Also

2006 SMT/Algebra Problems