document.write( "Question 34722This question is from textbook
\n" ); document.write( ": Prove with a simple equation that the product of four consecutive integers can never be a perfrct square. \n" ); document.write( "

Algebra.Com's Answer #20982 by longjonsilver(2297) $\"\"$ $\"About$

You can put this solution on YOUR website!
not sure if this is a rigourous proof, but it should at least show you some things.\r
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\n" ); document.write( "\n" ); document.write( "Take an example: 2*3*4*5. To be a perfect square, we need something like 3*3 or 7*7 or 15*15 etc.\r
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\n" ); document.write( "\n" ); document.write( "So with 2*3*4*5, we need two of the integers to multiply to equal the other two integers. The only remote possibility of this is if 2*5=3*4, which it doesn't in this case, since 10 is not 12.\r
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\n" ); document.write( "\n" ); document.write( "So, algebraically, x(x+1)(x+2)(x+3) are our four integers, starting at number \"x\".\r
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\n" ); document.write( "\n" ); document.write( "If we then say, OK, for which value of x does x(x+3) = (x+1)(x+2) hold? then we can find the answer(s). From the wording of the question, we are looking for no solution.\r
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\n" ); document.write( "\n" ); document.write( " $\"+x%28x%2B3%29+=+%28x%2B1%29%28x%2B2%29+\"$
\n" ); document.write( " $\"+x%5E2%2B3x+=+x%5E2+%2B+3x+%2B+2+\"$
\n" ); document.write( " $\"+0+=+2+\"$
\n" ); document.write( "which is patently not true, so there is no value of x such that $\"+x%28x%2B3%29+=+%28x%2B1%29%28x%2B2%29+\"$ is true.\r
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\n" ); document.write( "\n" ); document.write( "jon.
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