document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=392999>Question 392999</A>: <I><SPAN STYLE=\"word-break: break-all;\"> when a three digit number is divided by the sum of the digits of the number, the quotient is 26.  What is the least number for which this is true? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #278965 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=278965\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100a+%2B+10b+%2B+c%29%2F%28a+%2B+b+%2B+c%29+=+26 ALIGN=MIDDLE  ALT=\"%28100a+%2B+10b+%2B+c%29%2F%28a+%2B+b+%2B+c%29+=+26\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100a+%2B+10b+%2B+c+=+26a+%2B+26b+%2B+26c ALIGN=MIDDLE  ALT=\"100a+%2B+10b+%2B+c+=+26a+%2B+26b+%2B+26c\"   DISABLED_style= \"max-width:90%; height:auto;\" > (a, b, c are digits). Subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=99a+%2B+9b+=+25a+%2B+25b+%2B+25c ALIGN=MIDDLE  ALT=\"99a+%2B+9b+=+25a+%2B+25b+%2B+25c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This implies 9 divides a+b+c and 25 divides 11a+b. Therefore, the only possible values of a+b+c are 9, 18, 27 (since 36 is too high).\r<BR>\n" );
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document.write( "Suppose a+b+c = 9. Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb+=+9-c ALIGN=MIDDLE  ALT=\"a%2Bb+=+9-c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11a+%2B+b+=+9+-+c+%2B+10a ALIGN=MIDDLE  ALT=\"11a+%2B+b+=+9+-+c+%2B+10a\"   DISABLED_style= \"max-width:90%; height:auto;\" > (by substituting) which is divisible by 25. Since 9 - c + 10a is congruent to 9 - c (mod 10) = 5 or 0, the only possibility for c is 4 (since 9 is impossible). This leaves <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb+=+5 ALIGN=MIDDLE  ALT=\"a%2Bb+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note that 11a + b is divisible by 25). The easiest way is simply guess and check to find any solutions; we see that a = 2, b = 3 works since 11(2) + 3 is divisible by 25.\r<BR>\n" );
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document.write( "Therefore the smallest number is 234. </SPAN>\n" );
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