# 2002 AMC 12B Problems/Problem 4

The following problem is from both the 2002 AMC 12B #4 and 2002 AMC 10B #7, so both problems redirect to this page.

## Problem

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: $\mathrm{(A)}\ 2\ \text{divides\ }n \qquad\mathrm{(B)}\ 3\ \text{divides\ }n \qquad\mathrm{(C)}$ $\ 6\ \text{divides\ }n \qquad\mathrm{(D)}\ 7\ \text{divides\ }n \qquad\mathrm{(E)}\ n > 84$

## Solution 1

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 $\boxed{\mathrm{(E)}\ n>84}$.

## Solution 2 (no limits)

Since $\frac 12 + \frac 13 + \frac 17 = \frac {41}{42}$, it is very clear that $n=42$ makes the expression an integer. Because $n$ is a positive integer, $\frac{1}{n}$ must be less than or equal to $1$. Thus the only integer the expression can take is $1$, making the only value for $n$ $42$. Thus $\boxed{\mathrm{(E)}\ n>84}$

~superagh

## Solution 3(similiar to solution2)

Cross multiplying and adding the fraction we get the fraction to be equal to $\frac {41n + 42}{42n}$, This value has to be an integer. This implies, $42n|(41n+42)$.

=> $42|(41n + 42)$

  but $42|42$, hence $42|41n$
but $42$ does not divides $41$, $42|n$                                                                                                    -(1)


=> $n|(41n+42)$

  but $n|41n$, $n|42$                                                                                                    -(2)


from (1) and (2) we get that n=42. Comparing this with the options, we see that option E is the incorrect statement and hence E is the answer.

~rudolf1279

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 