Problem Log - 2023 IOQM No. 9

by Nibir Sankar, Jun 4, 2025

Problem

Find the number of triples $(a, b, c)$ of positive integers such that:

(a) $a b$ is a prime; (b) $b c$ is a product of two primes; © $a b c$ is not divisible by square of any prime and (d) $a b c \leq 30$

From (a), we know that one of $a$ or $b$ must be a prime and the other one 1.

Let’s say we pick $a = 1$ .

In this case, $b$ and $c$ must both be different primes (that is, $b \neq c$ , and both primes).

We find $A = {(a, b, c) : a b c \leq 30 ⟹ b c \leq 30}$ ( $∵ a = 1$ )

The set $A$ becomes:

\begin{matrix} {(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)} \end{matrix}

and $n (A) = 14$ .

Now we pick $b = 1$ .

In this case, since $b c$ is a product of two prime, and $b = 1$ , $c$ must be a product of two primes. That is $c \in {6, 10, 14, 15, \dots}$

We are also limited by $a$ , which is also a prime ( $∵ a b$ is a prime)

Now, we need to find $B = {(a, b, c) : a b c \leq 30 ⟹ a c \leq 30}$ ( $∵ b = 1$ )

Again, $∵ a b c$ should not be divisible by square of any prime, we need to make sure that $a c$ is not divisible by any square as well.

So, the set $B$ becomes:

\begin{array}{l} {(2, 1, 15), (5, 1, 6), (3, 1, 10)} \end{array}

and $n (B) = 3$ .

So, total possibilities = $n (A) + n (B) = 14 + 3 = 17$