2023 IOQM/Problem 9

Problem

Find the number of triples $(a, b, c)$ of positive integers such that (a) $ab$ is a prime;

(b) $bc$ is a product of two primes;

(c) $abc$ is not divisible by square of any prime and

(d) $abc\leq30$

Since, ab is a prime, this means that one of a and b is 1 and the other is prime. So, there are 2 cases from here:

Case 1(a=1)

If a is one and b is a prime, this means that c is also a prime as bc is a product of 2 primes.

Now, $abc\leq30$ ⇒ $bc\leq30$ , so all