Let n be a positive integer such that 12+13+17+1n is an integer. Which of the following statements is not true:

2002 AMC 12B Problem 4

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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

\dfrac12+\dfrac13+\dfrac17=\dfrac{41}{42}

The whole sum lies strictly between

0

and

2,

so it must equal

1

; solve for

n

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.