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 $\le$ 30.

Step 1: ab is prime mean one of them is 1.

Step 2: abc  {4, 9, 25}

Step 3: Case I : $a$ = 1 => $b$ is prime and bc prime product

Step 4: => $c$ is prime

Step 5: => abc \le 30

Step 6: => bc \le 30 & bc  {4, 9, 25}

Step 7: => (b, c) are different prime

Step 8: => $b$ = {2, 3, 5, 7}

Step 9: c = {3, 5, 7} and $b$ \neq c. Then total number of cases = 14.

Step 10: Case II : When $b$ = 1 then possible values of (a, b, c) are (2, 1, 15), (3, 1, 10) and (5, 1, 6).

Step 11: Total number of ways = 14 + 3 = 17