document.write( "Question 802626: Number of positive integer solutions for equation 2x +3y = 763 \n" ); document.write( "
Algebra.Com's Answer #483956 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "The largest coefficient in absolute value in 2x + 3y = 763 is 3 \r\n" );
document.write( "\r\n" );
document.write( "The nearest multiple of 3 to 763 is 765.\r\n" );
document.write( "\r\n" );
document.write( "763 = 765 - 2\r\n" );
document.write( "\r\n" );
document.write( "The LCM of both 2 and 3 is 6.  So we add and subtract a multiple of 6,\r\n" );
document.write( "namely 6n:\r\n" );
document.write( "\r\n" );
document.write( "763 = 765 - 2 + 6n - 6n = 6n - 2 + 765 - 6n = 2(3n-1) + 3(255-2n)\r\n" );
document.write( "\r\n" );
document.write( "Thus \r\n" );
document.write( "\r\n" );
document.write( "2(3n-1) + 3(255-2n) = 763  and x = 3n-1 and y = 255-2n\r\n" );
document.write( "\r\n" );
document.write( "Since y > 0,\r\n" );
document.write( "\r\n" );
document.write( "255-2n > 0\r\n" );
document.write( "   -2n > -255\r\n" );
document.write( "     n < 127.5\r\n" );
document.write( "\r\n" );
document.write( "So n can range from 1 to 127 inclusively.\r\n" );
document.write( "\r\n" );
document.write( "Therefore there are 127 solutions in positive integers.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );