NMOS 2011 Special Round question 2

There are only $8$ primes less than $20$ , so we use all of them. Their total is $77$ ; we have that $A=\frac{77-p}{p}$ for one of the primes $p<20$ . Since $A$ is an integer, $p$ must divide $77$ ; there are only two options, $p=7$ or $p=11$ , and to maximise $A$ we choose the smaller value $p=7$ giving $A=\boxed{10}$ .