AMC10 2002 B

AMC10 2002 B · Q7

AMC10 2002 B · Q7. It mainly tests Fractions, GCD & LCM.

Let $n$ be a positive integer such that $\frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{n}$ is an integer. Which of the following statements is not true:

设 $n$ 是正整数，使得 $\frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{n}$ 是整数。以下哪个陈述不正确：

(A) 2 divides $n$ 2 整除 $n$

(B) 3 divides $n$ 3 整除 $n$

(D) 7 divides $n$ 7 整除 $n$

(E) $n > 84$ $n > 84$

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) The number $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}$ is greater than 0 and less than $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{1}<2$. Hence, $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}=\frac{41}{42}+\frac{1}{n}$ is an integer precisely when it is equal to 1. This implies that $n=42$, so the answer is (E).

（E）数 $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}$ 大于 0 且小于 $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{1}<2$。因此， $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}=\frac{41}{42}+\frac{1}{n}$ 恰好在其等于 1 时为整数。这推出 $n=42$，所以答案是（E）。

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