1985 AMC 12 第 26 题

Question

求最小正整数

n

，使

\frac{n-13}{5n+6}

为非零可约分数。

Find the least positive integer

n

for which

\frac{n-13}{5n+6}

is a non-zero reducible fraction.

45

68

155

226

以上都不是

none of these

n-13

和

5n+6

的任何公因数也整除它们的一个适当线性组合

Any common divisor of

n-13

and

5n+6

also divides a suitable linear combination

计算

(5n+6)-5(n-13)

Compute

(5n+6)-5(n-13)

欧几里得算法给出

\begin{aligned} &\gcd(n-13,5n+6)\\ &\quad=\gcd(n-13,71)\text{。} \end{aligned}

由于

71

是素数，该非零分数可约当且仅当

n-13

是

71

的非零倍数。最小的正数情形为

n=13+71=84

，它不在前四个数值选项中。

所以正确答案是 E。

Euclid’s algorithm gives

\begin{aligned} &\gcd(n-13,5n+6)\\ &\quad=\gcd(n-13,71). \end{aligned}

Since

71

is prime, the nonzero fraction is reducible exactly when

n-13

is a nonzero multiple of

71.

The least positive possibility is

n=13+71=84,

which is not among the four numerical choices.

Thus the correct answer is E.

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