2023 IOQM/Problem 9
Problem
Find the number of triples of positive integers such that
(a)
is a prime;
(b) is a product of two primes;
(c) is not divisible by square of any prime and
(d)
Solution1(Casework)
Since, is a prime, this means that one of
and
is 1 and the other is prime. So, there are 2 cases from here:
Case 1()
If is one and
is a prime, this means that
is also a prime but different from
( as
is a product of 2 primes but
is not divisible by the square of any prime)
Now, ⇒
, so all possible pairs of
here are
Total no. of ordered pairs = 14 here
Case 2()
If is one and
is a prime, this means that
is the product of 2 different primes ( as
is a product of 2 primes but
is not divisible by the square of any prime)
Now, ⇒
also,
is not divisible by the square of any prime so a should not divide
so all possible pairs of
here are
Total no. of ordered pairs = 3 here
Hence, total no. of triplets = 14+3=
~SANSGANKRSNGUPTA