HMMT 2016 U26

For positive integers $a, b, a \uparrow \uparrow b$ is defined as follows: $a \uparrow \uparrow 1=a$, and $a \uparrow \uparrow b=a^{a \uparrow \uparrow(b-1)}$ if $b>1$.

Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow \uparrow 6 \not \equiv a \uparrow \uparrow 7$ $\bmod n$.