Question 1037770
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?<pre><b>75&#8729;n&#8729;p = (3&#8729;5&#8729;5)&#8729;n&#8729;p = 3&#8729;5&#8729;(5&#8729;n)&#8729;p = 

[5&#8729;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&#8729;n&#8729;p = 3&#8729;5&#8729;(5&#8729;5)&#8729;p = 3&#8729;(5&#8729;5&#8729;5)&#8729;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&#8729;3]

75&#8729;n&#8729;p = 3&#8729;(5&#8729;5&#8729;5)&#8729;p = 3&#8729;(5&#8729;5&#8729;5)&#8729;(3&#8729;3) = (3&#8729;3&#8729;3)(5&#8729;5&#8729;5) = 3³&#8729;5³

So n = 5, p = 3&#8729;3 = 3² = 9

Checking:  5&#8729;n = 5&#8729;5 = 25 = 5² 
          75&#8729;n&#8729;p = 75&#8729;5&#8729;9 = 3375 = 15³

n+p = 5+9 = 14.

Edwin</pre></b>