document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=543190>Question 543190</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "     Kindly provide the solution for this below mentioned problem?\r<BR>\n" );
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document.write( "Find the largest 5-digit number which when divided by 84, 72 and 96 leaves remainder 82, 70, and 94 respectively.\r<BR>\n" );
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document.write( "Regards<BR>\n" );
document.write( "Gopinath  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354786 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=354786\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The key to solving this problem is to realize that your mystery number is 2 short of a multiple of 84, 72, and 96. If we call your mystery number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B2 ALIGN=MIDDLE  ALT=\"n%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a multiple of 84, 72, and 96.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=84=7%2A12=2%5E2%2A3%2A7 ALIGN=MIDDLE  ALT=\"84=7%2A12=2%5E2%2A3%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=8%2A9=2%5E3%2A3%5E2 ALIGN=MIDDLE  ALT=\"72=8%2A9=2%5E3%2A3%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=96=32%2A3=2%5E5%2A3%5E2 ALIGN=MIDDLE  ALT=\"96=32%2A3=2%5E5%2A3%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The least common multiple of 84, 72, and 96 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LCM%2884%2C72%2C96%29=2%5E5%2A3%5E3%2A7=96%2A7=672 ALIGN=MIDDLE  ALT=\"LCM%2884%2C72%2C96%29=2%5E5%2A3%5E3%2A7=96%2A7=672\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the problem just asked for any \"number which when divided by 84, 72 and 96 leaves remainder 82, 70, and 94 respectively,\" we would say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B2=672 ALIGN=MIDDLE  ALT=\"n%2B2=672\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=670 ALIGN=MIDDLE  ALT=\"n=670\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be one of many solutions. All the solutions would be numbers such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B2=672%2Ak ALIGN=MIDDLE  ALT=\"n%2B2=672%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some positive integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "As the problem asks for \"the largest 5-digit number,\" we need to look for an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B2 ALIGN=MIDDLE  ALT=\"n%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiple of 672 that makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > the largest 5-digit number complying with the condition.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E5%2F672=about148.8 ALIGN=MIDDLE  ALT=\"10%5E5%2F672=about148.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > according to my calculator, so I can easily estimate that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B2=148%2A672 ALIGN=MIDDLE  ALT=\"n%2B2=148%2A672\"   DISABLED_style= \"max-width:90%; height:auto;\" > will yield a 5-digit <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while the next larger multiple of 672 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B2=149%2A672 ALIGN=MIDDLE  ALT=\"n%2B2=149%2A672\"   DISABLED_style= \"max-width:90%; height:auto;\" > will yield a 6-digit <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=148%2A672-2=99454 ALIGN=MIDDLE  ALT=\"n=148%2A672-2=99454\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the solution. </SPAN>\n" );
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