document.write( "Question 1136491: If d is the H.C.F of 45 and 27, find x, y satisfying d = 45x + 27y \n" ); document.write( "

Algebra.Com's Answer #754249 by ikleyn(52776) $\"\"$ $\"About$

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document.write( "H.C.F. is the Highest Common Factor and is the same as the Greatest Common Factor (GCF).\r\n" );
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document.write( "GCF of numbers 45 and 27 is 9, so the problem asks to find integers x and y satisfying\r\n" );
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document.write( "    9 = 45x + 27y.\r\n" );
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document.write( "Reducing by the factor 9 both sides, it is the same as to find integers x and y satisfying\r\n" );
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document.write( "    1 = 5x + 3y.\r\n" );
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document.write( "The solution is easy to find by the \"trial and error method\", and there is a nice method to automatize it using Excel.\r\n" );
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document.write( "You write y = ,  give x the series of values 1, 2, 3, . . .  and look for x which gives integer value for y (see the Table below):\r\n" );
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document.write( "  T A B L E\r\n" );
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document.write( "x        y\r\n" );
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document.write( "1	-1.333333333\r\n" );
document.write( "2	-3\r\n" );
document.write( "3	-4.666666667\r\n" );
document.write( "4	-6.333333333\r\n" );
document.write( "5	-8\r\n" );
document.write( "6	-9.666666667\r\n" );
document.write( "7	-11.33333333\r\n" );
document.write( "8	-13\r\n" );
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document.write( "In our case, the second line of the Table just gives the solution  x= 2  and  y= -3.\r\n" );
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document.write( "Notice that after that, every third line of the Table gives the next solution.\r\n" );
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document.write( "So, the answer for the original problem are these two values  x= 2,  y= -3.\r\n" );
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