document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=549506>Question 549506</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the remainder when 2^1024 + 5^1024 +1 is divided by 9? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #358040 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=358040\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Several ways to do this. One way is to that\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2^{1024} = 2(8^{341})\">, and that\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2(8^{341}) \equiv 2((-1)^{341}) \equiv 2(-1) \equiv 7\"> (modulo 9).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Similarly,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 5^{1024} = (5^4)(5^6)^{170} \equiv (5^4)(1^{170}) \equiv 5^4 \equiv 4\"> (modulo 9).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hence the sum is equivalent to 7 + 4 + 1 = 12 = 3 (modulo 9), so the remainder is 3. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );