For how many real values of x is √120 - √x an integer?

2006 AMC 10A Problem 10

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10.

For how many real values of

x

\sqrt{120 - \sqrt{x}}

an integer?

3

6

9

10

11

Let

k = \sqrt{120 - \sqrt{x}}

and bound

k

k

is an integer with

0 \le k \le 10

Let

k = \sqrt{120 - \sqrt{x}}.

Since

\sqrt{x} \ge 0,

we need

0 \le k \le \sqrt{120},

k \in \{0, 1, \ldots, 10\},

giving

11

values.

Each

k

yields

\sqrt{x} = 120 - k^2,

and since

120 - k^2

is positive and strictly decreasing, the resulting values

x = (120 - k^2)^2

are distinct.

Thus, the correct answer is E.