document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=563311>Question 563311</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Your task is to find two different numbers that sum to 1000 . One number must be a multiple of 19 and the number must be a multiple of 47 . Show or explain how you got your answer . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #364957 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=364957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 19a + 47b = 1000\r<BR>\n" );
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document.write( "Solving for a,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a = \frac{1000 - 47b}{19}\">\r<BR>\n" );
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document.write( "We want 1000-47b to be congruent to 0 mod 19. Since 1000 is 12 mod 19, 47b must be 12 mod 19. Also, 47 is congruent to 9 (mod 19), so we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 9b \equiv 12(mod 19)\">\r<BR>\n" );
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document.write( "If b = 14, then 9b = 126 which is 12 mod 19, so this works. Hence 47b = 47*14 = 658, 19*18 = 342. If you know Diophantine equations you can use this solution to generate infinitely many solutions. </SPAN>\n" );
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