AMC12 2002 B

AMC12 2002 B · Q4

AMC12 2002 B · Q4. It mainly tests Fractions.

Let $n$ be a positive integer such that $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ is an integer. Which of the following statements is not true:

设 $n$ 是正整数，使得 $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ 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

Since $\frac 12 + \frac 13 + \frac 17 = \frac {41}{42}$, $0 < \lim_{n \rightarrow \infty} \left(\frac{41}{42} + \frac{1}{n}\right) < \frac {41}{42} + \frac 1n < \frac{41}{42} + \frac 11 < 2$ From which it follows that $\frac{41}{42} + \frac 1n = 1$ and $n = 42$. The only answer choice that is not true is $\boxed{\mathrm{(E)}\ n>84}$.

由于 $\frac 12 + \frac 13 + \frac 17 = \frac {41}{42}$，且 $0 < \lim_{n \rightarrow \infty} \left(\frac{41}{42} + \frac{1}{n}\right) < \frac {41}{42} + \frac 1n < \frac{41}{42} + \frac 11 < 2$ 由此可得 $\frac{41}{42} + \frac 1n = 1$，并且 $n = 42$。唯一不正确的选项是 $\boxed{\mathrm{(E)}\ n>84}$。

Topics

AR.001 Fractions