document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019656>Question 1019656</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solving linear diophantine equation where c is greatly larger. (ax+by=c)\r<BR>\n" );
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document.write( "3x + 4y = 478\r<BR>\n" );
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document.write( "This seems pretty impossible to find the x and y using euclids algorithm..\r<BR>\n" );
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document.write( "Is it possible?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #635616 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=635616\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "Solving linear diophantine equation where c is greatly larger. (ax+by=c)<BR>\n" );
document.write( "3x + 4y = 478<BR>\n" );
document.write( "This seems pretty impossible to find the x and y using euclids algorithm..<BR>\n" );
document.write( "Is it possible?<BR>\n" );
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document.write( "Solution: <BR>\n" );
document.write( "Since GCF(3,4)=1, and 1|478, yes, there are infinite solutions.<BR>\n" );
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document.write( "One possible solution by inspection is x=-478, y=478, which gives in general<BR>\n" );
document.write( "x=-478+4k<BR>\n" );
document.write( "y=478-3k<BR>\n" );
document.write( "We see that k&ge;478/4=120 and k&le;478/3=159 for both x and y to be non-negative.<BR>\n" );
document.write( "Hence the valid values for x and y are 120&le;k&le;159.<BR>\n" );
document.write( "For example,<BR>\n" );
document.write( "k=120, x=2, y=118<BR>\n" );
document.write( "k=121, x=6, y=115<BR>\n" );
document.write( "k=122, x=10, y=112<BR>\n" );
document.write( "...<BR>\n" );
document.write( "k=159, x=158, y=1<BR>\n" );
document.write( "are all solutions to the given problem.<BR>\n" );
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document.write( "For a more detailed explanation, see answer to problem 1019647, or<BR>\n" );
document.write( "http://math.stackexchange.com/questions/20717/how-to-find-solutions-of-linear-diophantine-ax-by-c </SPAN>\n" );
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