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) $abc\leq30$

Solution1(Casework)

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 but different from b ( as bc is a product of 2 primes but abc is not divisible by the square of any prime)

Now, $abc\leq30$ ⇒ $bc\leq30$ , so all possible pairs of $(a,b,c)$ here are $(1,2,3); (1,2,5); (1,2,7); (1,2,11); (1,2,13); (1,3,2); (1,3,5); (1,3,7); (1,5,2); (1,5,3); (1,7,2); (1,7,3); (1,11,2); (1,13,2)$ Total no. of ordered pairs = 14 here

Case 2(b=1)

If b is one and $a$ is a prime, this means that c is the product of 2 different primes ( as bc is a product of 2 primes but abc is not divisible by the square of any prime)

Now, $abc\leq30$ ⇒ $ac\leq30$ also, abc is not divisible by the square of any prime so a should not divide c so all possible pairs of $(a,b,c)$ here are $(2,1,15); (3,1,10); (5,1,6)$ Total no. of ordered pairs = 3 here

Hence, total no. of (a,b,c) are 17 here. So the answer is $\boxed{17}$

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2023 IOQM/Problem 9

Problem

Solution1(Casework)