By looking at b and c together, we can deduce that 5 is the prime number in the mystery number's factorization, so we know our number is a product of 5 and one or more non-primes. Since the non-primes must be even numbers, and we know the factors can only contain ONE prime, the remaining factors MUST be multiples of 2, so our number is 5*2^x such that 5*2^x > 50 and 5*2^x < 100:
5*2*2*2*2 = 80
Our mystery number is 80!
I hope this helps!
