# Difference between revisions of "1985 AHSME Problems/Problem 26"

## Problem

Find the least positive integer $n$ for which $\frac{n-13}{5n+6}$ is a non-zero reducible fraction.

$\mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text{none of these}$

## Solution

For the fraction to be reducible, the greatest common factor of the numerator and the denominator must be greater than $1$. By the Euclidean algorithm,

$\gcd(5n+6, n-13)$

$\gcd(5n+6-5(n-13), n-13)$

$\gcd(71, n-13)$

Since $71$ is prime, $n-13$ must be a multiple of $71$, which first occurs when $n=71+13=84, \boxed{\text{E}}$.

## See Also

 1985 AHSME (Problems • Answer Key • Resources) Preceded byProblem 25 Followed byProblem 27 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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