document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=591789>Question 591789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the smallest odd number you can obtain from the product <BR>\n" );
document.write( "of four different prime numbers? Show or explain how you got<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #375730 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=375730\">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 product of any two odd numbers is odd:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (2m\ -\ 1)(2n\ -\ 1)\ =\ 4mn\ -\ 2m\ -\ 2n\ +\ 1\">.\r<BR>\n" );
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document.write( "Since\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\,|\,4mn\ -\ 2m\ -\ 2n\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4mn\ -\ 2m\ -\ 2n\">\r<BR>\n" );
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document.write( "is even, hence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4mn\ -\ 2m\ -\ 2n\ +\ 1\">\r<BR>\n" );
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document.write( "is odd.\r<BR>\n" );
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document.write( "Any product that has a factor of 2 is even, so in order to get 4 prime factors of an odd number, 2 must be excluded.  The smallest product is therefore obtained from the next four prime numbers.\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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