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 $n$

$n\gt84$

Answer: E

Concepts:fraction divisibility bounding to limit cases

Difficulty rating: 1270

Solution:

Since $\dfrac12+\dfrac13+\dfrac17=\dfrac{41}{42},$ the sum $\dfrac{41}{42}+\dfrac1n$ lies strictly between $0$ and $2,$ so it must equal $1.$ Then $\dfrac1n=\dfrac1{42},$ giving $n=42.$

Now $2,$ $3,$ $6,$ and $7$ all divide $42,$ but $n\gt84$ is false. The untrue statement is $n\gt84.$

Thus, the correct answer is E.

2002 AMC 12B Problem 4

Answer: E

Difficulty rating: 1270

Solution:

Problem 4 in Other Years