document.write( "Question 1205493: Solve : x! × y! = 120 × (xy)! \n" ); document.write( "

Algebra.Com's Answer #842341 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Solve : x! × y! = 120 × (xy)!
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "There are two solutions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "One solution is     x= 5, y= 0.   \r\n" );
document.write( "\r\n" );
document.write( "Other solution is   x= 0, y= 5.   \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let's prove first that  x= 5, y= 0 is the solution.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Indeed, then  x! = 5! = 120;\r\n" );
document.write( "\r\n" );
document.write( "              y! = 0! = 1   ( by the definition of 0! )\r\n" );
document.write( "\r\n" );
document.write( "        thus left side is  x! * y! = 120 * 1 = 120.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, for the right side  (xy)! = (5*0)! = 0! = 1 ( again, by the definition of 0! ),\r\n" );
document.write( "\r\n" );
document.write( "and thus left side, 120, is equal to the right side, which is 120, too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For other solution, x= 0, y= 5,  the proof is symmetric.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Edwin, \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "thanks for this valuable addition.\r
\n" ); document.write( "\n" ); document.write( "Happy new year!\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );