2008 AMC 12A Problem 5

Question

Suppose that

\dfrac{2x}{3} - \dfrac{x}{6}

is an integer. Which of the following statements must be true about

x?

It is negative.

It is even, but not necessarily a multiple of

3.

It is a multiple of

3,

but not necessarily even.

It is a multiple of

6,

but not necessarily a multiple of

12.

It is a multiple of

12.

Combine the two fractions over a common denominator

Simplify to

\dfrac{x}{2},

then decide what divisibility condition makes this an integer

Combining the fractions,

\dfrac{2x}{3} - \dfrac{x}{6} = \dfrac{4x - x}{6} = \dfrac{x}{2}.

This is an integer exactly when

x

is even. The example

x = 4

is even but not a multiple of

3,

which rules out every other statement.

Thus, B is the correct answer.

Problem 5 in other years