2002 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2002 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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Concepts:fraction divisibility bounding to limit cases

Difficulty rating: 1270

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 $n$

$n\gt84$

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.

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