Let $a$ and $b$ be distinct positive divisors of $80.$ What is the smallest possible value of $ab$ which is not a divisor of $80?$
The positive divisors of $80$ are $1,2,4,5,8,10,16,20,40,80.$ As $80=2^4\cdot 5$, the immediate possibilities for $ab$ that do not divide $80$ are $8\cdot 4=16\cdot 2=32$ and $5\cdot 10=50.$ As $32<50,$ the smallest $ab$ not a divisor of $80$ is $\boxed{32}.$