document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=465888>Question 465888</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the smallest number that increases its value by 12 when the number's order of the digits is reversed? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #319302 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=319302\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I think I've already solved this problem; there is no such positive integer. This is because the digit sum remains the same, so the remainder upon dividing by 9 will be the same.\r<BR>\n" );
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document.write( "To see this, take some random numbers like 583, 79, and 1056. Reversing their digits we get 385, 97, and 6501. If we take their differences we get\r<BR>\n" );
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document.write( "583 - 385 = 198 = 22*9<BR>\n" );
document.write( "79 - 97 = -18 = -2*9<BR>\n" );
document.write( "1056 - 6501 = -5445 = -605*9\r<BR>\n" );
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document.write( "In all these cases we are essentially adding or subtracting multiples of 9, and since 12 is not a multiple of 9 we cannot make a number increase by 12.\r<BR>\n" );
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document.write( "However this is entirely possible in other bases. If we have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?12_{13}\"> (12 in base 13 = 15) and we reverse the digits we get <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?21_{13}\">, and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 21_{13} - 12_{13} = 27_{10} - 15_{10} = 12_{10}\">\r<BR>\n" );
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document.write( "See if you can find the smallest base-10 number that, when written in a different base, the property holds (here the base-10 number was 15). </SPAN>\n" );
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