User:Bxiao31415
AOPS Contributions
Observe that if 
such that n is a solution to the desired equation, so is 
, where m is an integer, 
.
\\So we only need to consider n from 1 to 60.
As shown in Solution 2, there are 4 cases which we will split into 2 main cases:
\\
\\Case 1: 
or 
, 
\\Case 2: 
or 
, 
\\There are 4 values of n where 
satisfying or .
\\
\\I claim that there are 4 values of 
satisfying Case 1. Suppose x is one value of n satisfying or , and 
.
\\Hence the solutions satisfying or , are of the form 
, so the values of 
are 
(mod 5), so 
(mod 5) and hence the value of m is unique since 
to satisfy and 2 and 5 are relatively prime.
\\
\\A similar approach can be used to show the same for Case 2, that there are 4 values of .
\\
\\Hence our answer is 
.
