document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=517510>Question 517510</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Proposition: If 'a' is a type 1 unteger and 'b' is a type 2 integer, then (a^2-b) is a type 2 integer.\r<BR>\n" );
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document.write( "Note:a type 1 integer is defined as x=3y+1 and i type two integer is defined as x=3y+2\r<BR>\n" );
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document.write( "This is what have so far:\r<BR>\n" );
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document.write( "Proof: We let a be a type 1 integer and b be a type 2 integer. We will show that a^2-b is a type 2 integer. By the definitions of a type 1 integer and type 2 integer, there are integers m and n such that a=3m+1 and b=3n+2. By substitution and the use of algebra we see that a^2-b = (3m+1)2-(3n+2)<BR>\n" );
document.write( "                                                     = (9m2+6m+1-3n-2)<BR>\n" );
document.write( "                                                     = (9m2+6m-3n)-1-2<BR>\n" );
document.write( "                                                     = 3(3m2+2m-n)-1\r<BR>\n" );
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document.write( "I know that 3(3m2+2m-n)-1 is a type two integer, I just don't know how to re-write it so it satisfies the definition of a type 2 integer, meaning I don't know how to get it into the x=3y+2 format.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #344810 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=344810\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I presume by your definition that \"x is a type 1 integer if x = 3y + 1 for integer y\" and similarly for b. There are two ways to show this (the second solution is much easier and more elegant).\r<BR>\n" );
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document.write( "Solution 1: Simply expand the expression a^2 - b, e.g.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (3m+1)^2 - (3n+2)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = 9m^2 + 6m + 1 - 3n - 2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = 3(3m^2 + 2m - n) - 1\">\r<BR>\n" );
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document.write( "We can write it in the form of a \"type two\" integer by adding 3 then subtracting 3:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = 3(3m^2 + 2m - n) - 3 + 3 - 1\">\r<BR>\n" );
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document.write( "The -3 can be factored along with 3(3m^2 + 2m - n):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = 3(m^2 + 2m - n - 1) + 3 - 1 = 3(m^2 + 2m - n - 1) + 2\">, hence it is a type 2 integer. &#8718;<BR>\n" );
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document.write( "Solution 2: This involves modular arithmetic (http://en.wikipedia.org/wiki/Modular_arithmetic). It is clear that all integers congruent to 1 mod 3 are type 1 integers, and all integers congruent to 2 mod 3 are type 2 integers. Also,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a \equiv 1 (mod 3), b \equiv 2 (mod 3)\"> so it follows that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2 - b \equiv 1^2 - 2 \equiv -1 \equiv 2 (mod 3)\">\r<BR>\n" );
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document.write( "Therefore a^2 - b is a type 2 integer. &#8718; </SPAN>\n" );
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