document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=192950>Question 192950</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 31z5 is a multiple of 9, where z is a digit, what is the value of z? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #144804 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=144804\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If 31z5 is a multiple of 9 (where z is a single digit), then the digits MUST add to a number that is a multiple of 9 (hopefully, they will add to 9 itself).\r<BR>\n" );
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document.write( "So the digits must add to 0, 9, 18, 27, 36, 45, 54, 63, 72, etc...\r<BR>\n" );
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document.write( "So this gives us the equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1%2Bz%2B5=9k ALIGN=MIDDLE  ALT=\"3%2B1%2Bz%2B5=9k\"   DISABLED_style= \"max-width:90%; height:auto;\" > where \"k\" is an integer (ie whole number)\r<BR>\n" );
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document.write( "So if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the smallest possible k value where we get a solution), then... \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1%2Bz%2B5=9 ALIGN=MIDDLE  ALT=\"3%2B1%2Bz%2B5=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve for \"z\" to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%2B9=9 ALIGN=MIDDLE  ALT=\"z%2B9=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=0 ALIGN=MIDDLE  ALT=\"z=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=0 ALIGN=MIDDLE  ALT=\"z=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the number 3105\r<BR>\n" );
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document.write( "So 3105 is divisible by 9. Check: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3105%2F9=345 ALIGN=MIDDLE  ALT=\"3105%2F9=345\"   DISABLED_style= \"max-width:90%; height:auto;\" > remainder 0.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2 ALIGN=MIDDLE  ALT=\"k=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then... \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1%2Bz%2B5=18 ALIGN=MIDDLE  ALT=\"3%2B1%2Bz%2B5=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve for \"z\" to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%2B9=18 ALIGN=MIDDLE  ALT=\"z%2B9=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=9 ALIGN=MIDDLE  ALT=\"z=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=9 ALIGN=MIDDLE  ALT=\"z=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the number 3195\r<BR>\n" );
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document.write( "So 3195 is divisible by 9. Check: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3195%2F9=395 ALIGN=MIDDLE  ALT=\"3195%2F9=395\"   DISABLED_style= \"max-width:90%; height:auto;\" > remainder 0.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=3 ALIGN=MIDDLE  ALT=\"k=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B1%2Bz%2B5=27 ALIGN=MIDDLE  ALT=\"3%2B1%2Bz%2B5=27\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve for \"z\" to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%2B9=27 ALIGN=MIDDLE  ALT=\"z%2B9=27\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=18 ALIGN=MIDDLE  ALT=\"z=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But since \"z\" is a single digit, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=z%3C=9 ALIGN=MIDDLE  ALT=\"0%3C=z%3C=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So ANY value of \"k\" greater than 2 will give us a \"z\" value greater than 9. So this means that we're done looking for values of \"z\".\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the two values of \"z\" are 0 and 9 produce the multiples of 9 3105 and 3195 respectively. </SPAN>\n" );
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