Problem:
Find the least positive integer n for which 5n+6n−13​ is a non-zero reducible fraction.
Answer Choices:
A. 45
B. 68
C. 155
D. 226
E. none of these
Solution:
5n+6n−13​ is reducible and nonzero iff its reciprocal n−135n+6​ exists and is reducible. By long division, n−135n+6​=5+n−1371​. Thus it is necessary and sufficient that n−1371​ be reducible. Since 71 is a prime, n−13 must be a multiple of 71. So n−13=71, or n=84, is the smallest solution.
OR
We seek the smallest n>0 for which n−13 and 5n+6 have a common factor and n−13î€ =0. To make it easier to see a common factor, set m=n−13; then 5n+6=5m+71. Clearly, m and 5m+71 have a common factor iff m and 71 do. Since 71 is a prime, m must be one of …,−71,0, 71,142,…. Thus n must be one of …,−58,13,84,155,…. The smallest positive value of n giving a positive fraction is n=84.