Question

Let $n$ be a positive integer such that $\dfrac12 + \dfrac13 + \dfrac17 + \dfrac1n$ is an integer. Which of the following statements is not true?

$2$ divides $n$

$3$ divides $n$

$6$ divides $n$

$7$ divides $n$

$n \gt 84$

Answer: E

Concepts:fraction divisibility bounding to limit cases

Difficulty rating: 1170

Solution:

The sum $\dfrac12 + \dfrac13 + \dfrac17 + \dfrac1n$ is greater than $0$ and less than $\dfrac12 + \dfrac13 + \dfrac17 + 1 \lt 2,$ so as an integer it must equal $1.$

Since $\dfrac12 + \dfrac13 + \dfrac17 = \dfrac{41}{42},$ we need $\dfrac1n = \dfrac{1}{42},$ so $n = 42.$

Then $2,$ $3,$ $6,$ and $7$ all divide $42,$ but $n = 42$ is not greater than $84.$ So the false statement is $n \gt 84.$

Thus, the correct answer is E.

2002 AMC 10B Problem 7

Answer: E

Difficulty rating: 1170

Solution:

Problem 7 in Other Years