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 $n=71+13=84, \boxed{\text{E}}$ .

1985 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 25	Followed by Problem 27
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1985 AHSME Problems/Problem 26

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