2002 AMC 10B Problem 7

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

The sum is positive and less than

2,

so it must equal exactly

1

Solve

\dfrac1n = 1 - \dfrac12 - \dfrac13 - \dfrac17

for

n

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.

Problem 7 in other years