document.write( "Question 733043: Consecutive Integers\r
\n" ); document.write( "\n" ); document.write( "If abcd are positive consecutive integers then is a+b^2+c^3 is divisible by d? \r
\n" ); document.write( "\n" ); document.write( "If it must always be, prove it. (It does not have to be a formal proof, but you cannot just show me a lot of example.)\r
\n" ); document.write( "\n" ); document.write( "If it isnt always, show me an example where it isnt. \n" ); document.write( "

Algebra.Com's Answer #448234 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
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\n" ); document.write( "\n" ); document.write( "If

, and

are consecutive positive integers, such that

, and

, then\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If

is always divisible by

then

must have a zero remainder.\r
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\n" ); document.write( "\n" ); document.write( "Substitute:\r
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\n" ); document.write( "\n" ); document.write( "

\r
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\n" ); document.write( "\n" ); document.write( "Expand the numerator and collect like terms\r
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\n" ); document.write( "\n" ); document.write( "

\r
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\n" ); document.write( "\n" ); document.write( "

\r
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\n" ); document.write( "\n" ); document.write( "Use synthetic division
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\r\n" );
document.write( "\r\n" );
document.write( "-3  |  1    7    15    9\r\n" );
document.write( "    |      -3   -12   -9\r\n" );
document.write( "    --------------------\r\n" );
document.write( "       1    4     3    0\r\n" );
document.write( "

\r
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\n" ); document.write( "\n" ); document.write( "The remainder is zero, so

where

, and

are consecutive positive integers such that

, and

.\r
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\n" ); document.write( "\n" ); document.write( "John
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\n" ); document.write( "Egw to Beta kai to Sigma
\n" ); document.write( "My calculator said it, I believe it, that settles it
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$\"The$

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