document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=615553>Question 615553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help using modular arithmetic to find the last two digits of a big number, for example 142^291 or something. I know how to use modular arithmetic to find the units digit, but not the last two digits. I'm only a beginner at modular arithmetic, and would like to get better at it. \r<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 #387242 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=387242\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 142 = 42 (mod 100)\r<BR>\n" );
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document.write( "142^2 = 142*142 = 42*42 = 1764 = 64 (mod 100)\r<BR>\n" );
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document.write( "144^4 = (144^2)^2 = 64^2 = 4096 = 96 (mod 100)\r<BR>\n" );
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document.write( "144^8 = (144^4)^2 = 96^2 = 9216 = 16 (mod 100)\r<BR>\n" );
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document.write( "144^16 = (144^8)^2 = 16^2 = 256 = 56 (mod 100)\r<BR>\n" );
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document.write( "144^32 = (144^16)^2 = 56^2 = 3136 = 36 (mod 100)\r<BR>\n" );
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document.write( "144^64 = (144^32)^2 = 36^2 = 1296 = 96 (mod 100)\r<BR>\n" );
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document.write( "144^128 = (144^64)^2 = 96^2 = 9216 = 16 (mod 100) ... Notice that there's a pattern starting to emerge\r<BR>\n" );
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document.write( "144^256 = (144^128)^2 = 16^2 = 256 = 56 (mod 100)\r<BR>\n" );
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document.write( "So \r<BR>\n" );
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document.write( "142^2 = 64 (mod 100)\r<BR>\n" );
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document.write( "144^4 = 96 (mod 100)\r<BR>\n" );
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document.write( "144^8 = 16 (mod 100)\r<BR>\n" );
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document.write( "144^16 = 56 (mod 100)\r<BR>\n" );
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document.write( "144^32 = 36 (mod 100)\r<BR>\n" );
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document.write( "144^64 = 96 (mod 100)\r<BR>\n" );
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document.write( "144^128 = 16 (mod 100) \r<BR>\n" );
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document.write( "144^256 = 56 (mod 100)\r<BR>\n" );
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document.write( "142^291 (mod 100)\r<BR>\n" );
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document.write( "142^(256 + 32 + 2 + 1) (mod 100)\r<BR>\n" );
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document.write( "( 142^256 ) * ( 142^32 ) * ( 142^2 ) * ( 142^1 ) (mod 100)\r<BR>\n" );
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document.write( "( 56 ) * ( 36 ) * ( 64 ) * ( 42 ) (mod 100)\r<BR>\n" );
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document.write( "5419008 (mod 100)\r<BR>\n" );
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document.write( "8 (mod 100)\r<BR>\n" );
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document.write( "Therefore 142^291 = 8 (mod 100)\r<BR>\n" );
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document.write( "So the last two digits of 142^291 are <font color=\"red\">08</font> </SPAN>\n" );
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