document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208954>Question 1208954</A>: <I><SPAN STYLE=\"word-break: break-all;\"> not sure how to prove this?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if n is odd then n^2 = 1 (mod 4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847497 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847497\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "not sure how to prove this?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if n is odd then n^2 = 1 (mod 4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "thanks!<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is easy.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "If n is odd integer number, it can be presented in the form  n = 2k+1,  \r\n" );
document.write( "where k is some integer number.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then \r\n" );
document.write( "\r\n" );
document.write( "    n^2 = (2k)^2 + 2*(2k) + 1 = 4k^2 + 4k + 1.    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In the right side, first addend, 4k^2, is a multiple of 4, \r\n" );
document.write( "and the second addend 4k is a multiple of k.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, the sum of the first two addends is a multiple of k.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus n^2 = 1 (mod 4).\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "At this point, the proof is complete.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In whole, this all is at the level of self-evident truths.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );