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 $1$ . Using the Euclidean algorithm, we compute $\begin{align*}\gcd\left(5n+6,n-13\right) &= \gcd\left(5n+6-5(n-13),n-13\right) \\ &= \gcd\left(71,n-13\right).\end{align*}$ Since $71$ is prime, it follows that this GCD will be $1$ unless $(n-13)$ is a multiple of $71$ , which first occurs when $n = 71+13 = 84$ , so the answer is $\boxed{\text{(E) none of these}}$ .

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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