document.write( "Question 253049: If x is a positive integer, which of the following CANNOT be an integer?\r
\n" ); document.write( "\n" ); document.write( "(A) \"+sqrt%28x-1%29+\" (B) \"sqrt%28x%5E2+-+1%29\" (C) \"+%28x%2B4%29+%2F+%28x-2%29+\" (D) \"+%28x%2B4%29+%2F+%28x%2B2%29+\" (E) \"+5%2F%28x%2B1%29+\"
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Algebra.Com's Answer #845360 by greenestamps(13203)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Look for counterexamples for each answer choice -- that is, positive values of x for which the expression is an integer.

\n" ); document.write( "(A) \"+sqrt%28x-1%29+\"
\n" ); document.write( "x=1 yields value 0, which is an integer

\n" ); document.write( "(B) \"sqrt%28x%5E2+-+1%29\"
\n" ); document.write( "again x=1 yields value 0

\n" ); document.write( "(C) \"+%28x%2B4%29+%2F+%28x-2%29+\"
\n" ); document.write( "x=3 makes the denominator 1 and so yields an integer value

\n" ); document.write( "(E) \"+5%2F%28x%2B1%29+\"
\n" ); document.write( "x=4 yields value 1, which is an integer

\n" ); document.write( "(D) \"+%28x%2B4%29+%2F+%28x%2B2%29+\"

\n" ); document.write( "I left this for last, because (D) is the answer.

\n" ); document.write( "To see this, note that \"+%28x%2B4%29+%2F+%28x%2B2%29+=+1%2B2%2F%28x%2B2%29\"

\n" ); document.write( "In that reformatted expression, 1 is an integer, so \"2%2F%28x%2B2%29\" must be an integer. But \"2%2F%28x%2B2%29\" is an integer only when (x+2) is a factor of 2 -- i.e., when (x+2) is -2, -1, 1, or 2 -- and x is not a positive integer for any of those.

\n" ); document.write( "ANSWER: (D)

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