Problem:
Find the least positive integer n for which 5n+6nβ13β is a non-zero reducible fraction.
Answer Choices:
A. 45
B. 68
C. 155
D. 226
E. none of these
Solution:
5n+6nβ13β is reducible and nonzero iff its reciprocal nβ135n+6β exists and is reducible. By long division, nβ135n+6β=5+nβ1371β. Thus it is necessary and sufficient that nβ1371β be reducible. Since 71 is a prime, nβ13 must be a multiple of 71. So nβ13=71, or n=84, is the smallest solution.
OR
We seek the smallest n>0 for which nβ13 and 5n+6 have a common factor and nβ13ξ =0. To make it easier to see a common factor, set m=nβ13; 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 β¦,β71,0, 71,142,β¦. Thus n must be one of β¦,β58,13,84,155,β¦. The smallest positive value of n giving a positive fraction is n=84.