document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=592549>Question 592549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the sum of the digits of a two-digit prime number is subtracted from the number. Prove that the difference cannot be a prime nuber. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #375991 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=375991\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The fact that the original number is prime is just a red herring.  In fact, if you subtract the sum of the digits from any integer, the result is divisible by 9 and therefore not prime.\r<BR>\n" );
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document.write( "Proof for two-digit integers:\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> represent the 10s digit and let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> represent the 1s digit.\r<BR>\n" );
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document.write( "Then the original integer is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 10x\ +\ y\"> and the sum of the digits is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ +\ y\">.\r<BR>\n" );
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document.write( "Subtracting:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 10x\ +\ y\ -\ (x\ +\ y)\ =\ 9x\">\r<BR>\n" );
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document.write( "Which is divisible by 9.  So if ANY two digit integer less the sum of its digits is non-prime, any prime two digit integer must exhibit the same characteristic.\r<BR>\n" );
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document.write( "Q.E.D.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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