Difference between revisions of "2023 IOQM/Problem 9"

Revision as of 07:25, 20 October 2023

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

@@ Line 9: / Line 9: @@
 (d) <math>abc\leq30</math>
+==Solution==
+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, <math>abc\leq30</math> ⇒ <math>bc\leq30</math>, so all