75∙n∙p = (3∙5∙5)∙n∙p = 3∙5∙(5∙n)∙p = 

[5∙n must be the square of an integer, and that will be
the case if n contributes an odd number of factors of 5.
So we try n as just 1 factor of 5, i.e., n=5] 

75∙n∙p = 3∙5∙(5∙5)∙p = 3∙(5∙5∙5)∙p

[That has 3 factors of 5, and thus the whole thing will be the 
cube of an integer if p contributes 2 more factors of 3.
So we take p = 3∙3]

75∙n∙p = 3∙(5∙5∙5)∙p = 3∙(5∙5∙5)∙(3∙3) = (3∙3∙3)(5∙5∙5) = 3³∙5³

So n = 5, p = 3∙3 = 3² = 9

Checking:  5∙n = 5∙5 = 25 = 5² 
          75∙n∙p = 75∙5∙9 = 3375 = 15³

n+p = 5+9 = 14.

Edwin