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

2006 AMC 12A 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

Set

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

for a nonnegative integer

k

Need

k^2 \le 120,

and each valid

k

gives a distinct

x = (120 - k^2)^2

Let

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

be an integer. Then

k \ge 0

and

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

0 \le k \le 10.

Each such

k

gives

\sqrt{x} = 120 - k^2 \ge 0,

hence a distinct value

x = (120 - k^2)^2.

That is

11

values.

Thus, the correct answer is E.