document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=978706>Question 978706</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi tutors, can you help me answer this question? Thanks\r<BR>\n" );
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document.write( "A three-digit number has all digits odd. How many such numbers are divisible by three?\r<BR>\n" );
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document.write( "A. 29<BR>\n" );
document.write( "B. 36<BR>\n" );
document.write( "C. 39<BR>\n" );
document.write( "D. 40<BR>\n" );
document.write( "E. 41 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #600223 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=600223\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I count <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2841%29 ALIGN=MIDDLE  ALT=\"highlight%2841%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A5%2A5=125 ALIGN=MIDDLE  ALT=\"5%2A5%2A5=125\"   DISABLED_style= \"max-width:90%; height:auto;\" > three-digit numbers whose digits are all odd.<BR>\n" );
document.write( "I counted it through a shortcut, but it is still a messy, cumbersome way to count it.<BR>\n" );
document.write( "Maybe the kids at the artofproblemsolving forum would have an elegant, streamlined way to count them.\r<BR>\n" );
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document.write( "HOW I COUNT THEM:<BR>\n" );
document.write( "For a number to be divisible by 3 (multiple of 3), its digits must add to a multiple of 3.<BR>\n" );
document.write( "I count <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets of 3 digits with all digits are odd, and they add to<BR>\n" );
document.write( "a multiple of 3 (3, 6, 9, 12, 15, 18, 21, 24, or 27).<BR>\n" );
document.write( "Those sets are:<BR>\n" );
document.write( "{1,1,1} , {1,1,7} ,<BR>\n" );
document.write( "{1,3,5} , {1,5,9} ,<BR>\n" );
document.write( "{1,7,7} ,<BR>\n" );
document.write( "{3,3,3} , {3,3,9} ,<BR>\n" );
document.write( "{3,5,7} ,<BR>\n" );
document.write( "{3,9,9} ,<BR>\n" );
document.write( "{5,5,5} ,<BR>\n" );
document.write( "{5,7,9} ,<BR>\n" );
document.write( "{7,7,7} , and<BR>\n" );
document.write( "{9,9,9} .\r<BR>\n" );
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document.write( "Among those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets have 3 different digits,<BR>\n" );
document.write( "Each of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets can be arranged in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2=6 ALIGN=MIDDLE  ALT=\"3%2A2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > different arrangements/permutations,<BR>\n" );
document.write( "making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A6=24 ALIGN=MIDDLE  ALT=\"4%2A6=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > different three-digit numbers whose digits are all odd.\r<BR>\n" );
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document.write( "Among those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets listed above,<BR>\n" );
document.write( "there are also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets made of just one single repeated digit,<BR>\n" );
document.write( "and each one of those sets can be arranged just one way,<BR>\n" );
document.write( "to form just one three-digit number,<BR>\n" );
document.write( "so from them we can get another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > three-digit numbers whose digits are all odd.\r<BR>\n" );
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document.write( "The remaining <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets listed above contain only two different digits, one of them repeated.<BR>\n" );
document.write( "Form each of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets, we can make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > different three-digit numbers,<BR>\n" );
document.write( "because there are 3 positions to place the unrepeated digit,<BR>\n" );
document.write( "and that gives us another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A3=12 ALIGN=MIDDLE  ALT=\"4%2A3=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > three-digit numbers whose digits are all odd.\r<BR>\n" );
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document.write( "That makes a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2B5%2B12=highlight%2841%29 ALIGN=MIDDLE  ALT=\"24%2B5%2B12=highlight%2841%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > three-digit numbers divisible by 3, whose digits are all odd.<BR>\n" );
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