N and p are integers greater than one. 5n is the square of
a number. 75np is the cube of a number. What is the smallest
value for n + p ?
Method?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
