document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=577782>Question 577782</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a three digit number , the hundreds digit , the tens digit and the units digit are in descending order and in Arithmetic progression . Each digit of the number is multiplied by the sum of the other 2 digits . The sum of all such results is A. B is the product of the tens digit and the sum of all the digits and A = (4/3)B . Find the number of such numbers ..\r<BR>\n" );
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document.write( "Options available are \r<BR>\n" );
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document.write( "a)4  b)3  c)2  d) More than 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #371028 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=371028\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose the digits are a, a-d, a-2d. Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A = a(2a-3d) + (a-d)(2a-2d) + (a-2d)(2a-d)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = 6a^2 - 12ad + 4d^2\">\r<BR>\n" );
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document.write( "Additionally, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE B = (a-d)(3a-3d) = 3a^2 - 6ad + 3d^2\">. The problem tells us that A = (4/3)B so \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 6a^2 - 12ad + 4d^2 = \frac{4}{3}(3a^2 - 6ad + 3d^2) \Rightarrow 2a^2 - 4ad = 0\"> (upon working out all the algebra)\r<BR>\n" );
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document.write( "This factors nicely to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2a(a-2d) = 0\">\r<BR>\n" );
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document.write( "Either a = 0 or a-2d = 0 (a = 2d). a cannot equal 0 because we have a three digit number (either the digits are negative or the number is 000) so a = 2d. If this is the case, then the only possible numbers are\r<BR>\n" );
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document.write( "a=2, d=1 --> 210<BR>\n" );
document.write( "a=4, d=2 --> 420<BR>\n" );
document.write( "a=6, d=3 --> 630<BR>\n" );
document.write( "a=8, d=4 --> 840\r<BR>\n" );
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document.write( "4 possible numbers, answer is A. </SPAN>\n" );
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