1985 AHSME Problems/Problem 26

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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 $\boxed{\text{(E) none of these}}$.

See Also

1985 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 25
Followed by
Problem 27
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