SOLUTION: What is the least positive integral value of n for which (n - 12) divided by (5n + 23) is a non-zero reducible fraction?

Question 548989: What is the least positive integral value of n for which (n - 12) divided by
(5n + 23) is a non-zero reducible fraction?

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
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:

(mod p)

This implies

(mod p), so

(mod p). Hence, p = 83. Therefore, we can set n = 95, obtaining the fraction 83/498, or 1/6.

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