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| − | == Problem ==
| + | #redirect [[2006 AMC 12A Problems/Problem 10]] |
| − | For how many real values of <math>\displaystyle x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer?
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| − | <math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math>
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| − | == Solution ==
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| − | Since <math>\sqrt{x}</math> cannot be negative, the outermost [[radicand]] is at most 120. We are interested in the number of integer values that the expression can take, so we count the number of squares less than 120, the greatest of which is <math>10^2=100.</math>
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| − | Thus our set of values is
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| − | <center><math> \{10^2, 9^2,\ldots,2^2, 1^2, 0^2\} </math></center>
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| − | And our answer is '''11, (E)'''
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| − | == See Also ==
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| − | *[[2006 AMC 10A Problems]]
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| − | *[[2006 AMC 10A Problems/Problem 9|Previous Problem]]
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| − | *[[2006 AMC 10A Problems/Problem 11|Next Problem]]
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| − | [[Category:Introductory Algebra Problems]]
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