document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1040980>Question 1040980</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When Joyce counts the pennies in her bank by fives, she has one left over. When she counts them by threes, there are two left over. What is the least possible number of pennies in the bank? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #655933 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=655933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let N = the possible number of pennies of Joyce in the bank.\r<BR>\n" );
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document.write( "==> N = 5r + 1 for some non-negative integer r.<BR>\n" );
document.write( "==> N = 3s + 2 for some non-negative integer s.\r<BR>\n" );
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document.write( "==> 5r + 1 = 3s + 2 ===> 5r - 3s = 1, \r<BR>\n" );
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document.write( "which is a linear Diophantine equation.\r<BR>\n" );
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document.write( "Now any linear Diophantine equation \r<BR>\n" );
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document.write( "ar - bs = c \r<BR>\n" );
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document.write( "has an integer solution in r and s if and only if  
gcd
(a,b) divides c, and that all integer solutions are of the form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+r%5B0%5D+-+%28bk%29%2Fgcd%28a%2Cb%29 ALIGN=MIDDLE  ALT=\"r+=+r%5B0%5D+-+%28bk%29%2Fgcd%28a%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+s%5B0%5D+%2B+%28ak%29%2Fgcd%28a%2Cb%29 ALIGN=MIDDLE  ALT=\"s+=+s%5B0%5D+%2B+%28ak%29%2Fgcd%28a%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B0%5D ALIGN=MIDDLE  ALT=\"r%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B0%5D ALIGN=MIDDLE  ALT=\"s%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > are particular solutions.\r<BR>\n" );
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document.write( "gcd(5,3) = 1 divides c = 1 ==> there are general solutions for the DE equation.<BR>\n" );
document.write( "A particular solution to this equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B0%5D+=+2 ALIGN=MIDDLE  ALT=\"r%5B0%5D+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B0%5D+=+3 ALIGN=MIDDLE  ALT=\"s%5B0%5D+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> r = 2 + 3k and s = 3 + 5k,<BR>\n" );
document.write( "for k = 0, 1, 2, 3, 4, ...<BR>\n" );
document.write( "are the general solutions.\r<BR>\n" );
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document.write( "r = 2 and s = 3 (when k = 0) give the lowest possible value for N which is  \r<BR>\n" );
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document.write( "N = 11,\r<BR>\n" );
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document.write( "and therefore 11 is the least possible number of pennies in the bank. </SPAN>\n" );
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