The number R has exactly 7 different positive integer factors, the number S has exactly 8 different positive integer factors, and their product R ∙ S has exactly M different positive integer factors. Compute the sum of all different possible values of M.
R is a prime raised to the 6th power.
S is the product of 3 distinct primes,
the product of a prime and a 3rd power of a prime, or a 7th power of a prime.
The main issue is whether R’s prime is one of S’s.
If it is and corresponds to a 1st power prime of S, RS has 8⋅4=32 divisors.
If it is and corresponds to a 3rd power prime of S, RS has 2⋅10=20 divisors.
If it is and corresponds to a 7th power prime of S, RS has 14 divisors.
Otherwise, RS has 7⋅8=56 divisors.
Therefore the sum of M==[32 + 20 + 14 + 56]==122