document.write( "Question 1205778: Hi, I have shown my working for the question below, am I on the right track?\r
\n" ); document.write( "\n" ); document.write( "〖Simplify 10〗^241 mod(13) \r
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\n" ); document.write( "\n" ); document.write( "Fermat’s Little Theorem states:
\n" ); document.write( "a^(p-1)≡1mod(p)\r
\n" ); document.write( "\n" ); document.write( "Substitute:
\n" ); document.write( "10^12≡1mod(13) \r
\n" ); document.write( "\n" ); document.write( "By our multiplication theorem we know that if
\n" ); document.write( "10^12≡1mod(13)
\n" ); document.write( "Then
\n" ); document.write( "10^((12)A)≡(1)^A mod (13) \r
\n" ); document.write( "\n" ); document.write( "We want to get from a power of 12 up to around 241, and 12 x 20 = 240
\n" ); document.write( " 10^(12)(20) ≡1^20 mod(13)
\n" ); document.write( " ≡1mod(13)\r
\n" ); document.write( "\n" ); document.write( "So what we have so far is
\n" ); document.write( "10^241 mod(13)≡10^240×10^1 mod(13)
\n" ); document.write( " ≡1×10mod(13)
\n" ); document.write( " ≡10mod(13)\r
\n" ); document.write( "\n" ); document.write( "When 10^241is divided by 13, the remainder is 10 \r
\n" ); document.write( "\n" ); document.write( "Thank you \n" ); document.write( "
Algebra.Com's Answer #842813 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I'll elaborate a little more on math_tutor2020's 2nd approach.\r\n" );
document.write( "As he pointed out, even if you knew nothing about modular \r\n" );
document.write( "arithmetic, and never heard of Fermat's little (or last) \r\n" );
document.write( "theorem, you could do it with only long division, make an \r\n" );
document.write( "indefinitely long string power of 10 like 1000000000000... \r\n" );
document.write( "and start dividing \r\n" );
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document.write( "         769230     \r\n" );
document.write( "    13)1000000000000...\r\n" );
document.write( "        91\r\n" );
document.write( "         90\r\n" );
document.write( "         78\r\n" );
document.write( "         120    \r\n" );
document.write( "         117\r\n" );
document.write( "           30\r\n" );
document.write( "           26\r\n" );
document.write( "            40\r\n" );
document.write( "            39\r\n" );
document.write( "             10\r\n" );
document.write( "              0\r\n" );
document.write( "             10 \r\n" );
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document.write( "Now we see by the red remainders, that they are going to keep\r\n" );
document.write( "going 10,9,12,3,4,1,... forever.  That's a cycle of 6 remainders\r\n" );
document.write( "since 101 had the reminder 10, we can find out by\r\n" );
document.write( "division  \r\n" );
document.write( "           48\r\n" );
document.write( "        6)241\r\n" );
document.write( "          24\r\n" );
document.write( "            1\r\n" );
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document.write( "This shows the remainders would go through the cycle of 6\r\n" );
document.write( "48 times and since the remainder is 1, that means it had \r\n" );
document.write( "started through the cycle the 49th time and was at the 1st \r\n" );
document.write( "member of the cycle when it got to the 241st power of 10.\r\n" );
document.write( "So the answer is the first member of the cycle, or 10.\r\n" );
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document.write( "No doubt Mr. Fermat started out this way and kept looking\r\n" );
document.write( "for patterns, and even patterns of patterns, etc. and came\r\n" );
document.write( "up with his little theorem. \r\n" );
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document.write( "Edwin
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