Question 548989
Assume n > 12 so that the fraction is positive (1 through 11 don't work anyway). If the fraction is reducible for some n, then n-12 and 5n+23 must have a common factor p (other than 1). We can write this using modular arithmetic:


*[tex \LARGE n-12 \equiv 5n + 23 \equiv 0] (mod p)


This implies *[tex \LARGE n \equiv 12] (mod p), so *[tex \LARGE 5n + 23 \equiv 5(12) + 23 \equiv 83 \equiv 0] (mod p). Hence, p = 83. Therefore, we can set n = 95, obtaining the fraction 83/498, or 1/6.