Question 128
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 $b \gt 1$.
Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow \uparrow 6 \not\equiv a \uparrow \uparrow 7 \pmod{n}$.
Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow \uparrow 6 \not\equiv a \uparrow \uparrow 7 \pmod{n}$.