document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1067319>Question 1067319</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many 3 digit palindromes are divisible by 3? Thanks for any help . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #682520 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=682520\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> While the middle digit could be any of the 10 digits, including 0,<BR>\n" );
document.write( "he first and last digits cannot be zero.<BR>\n" );
document.write( "They must be one of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > choices from 1 though 9.<BR>\n" );
document.write( "So, there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2A10=90 ALIGN=MIDDLE  ALT=\"9%2A10=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > 3-digit palindromes,<BR>\n" );
document.write( "For a number to be divisible by 3 , the sum of the digits must be divisible by 3.<BR>\n" );
document.write( "Of the 9 choices for first (and last) digit, 3 are multiples of 3,<BR>\n" );
document.write( "another 3 have a remainder of 1 when divided by 3,<BR>\n" );
document.write( "and the other 3 have a remainder of 2.<BR>\n" );
document.write( "For any middle digit, there will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > choices of first and last digits that will make the sum of digits a multiple of 3.<BR>\n" );
document.write( "If the middle digit is has a remainder of 0 when divided by 3 (as happens for 0, 3, or 9),<BR>\n" );
document.write( "a first/last digit that is a multiple of 3 makes the sum a multiple of 3.<BR>\n" );
document.write( "If you the middle digit has a remainder of 1 when divided by 3,<BR>\n" );
document.write( "a first/last digit with a reminder of 1,<BR>\n" );
document.write( "will make the 3 remainders add up to 3,<BR>\n" );
document.write( "for a sum that is a multiple of 3.<BR>\n" );
document.write( "If you the middle digit has a remainder of 2 when divided by 3,<BR>\n" );
document.write( "a first/last digit with a reminder of 2,<BR>\n" );
document.write( "will make the 3 remainders add up to 6,<BR>\n" );
document.write( "for a sum that is a multiple of 3.<BR>\n" );
document.write( "Of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" > palindromes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%2F3=highlight%2830%29 ALIGN=MIDDLE  ALT=\"90%2F3=highlight%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "will be divisible by 3. </SPAN>\n" );
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