document.write( "Question 1207676: Find the remainder when 40^{13} is divided by 81. \n" ); document.write( "

Algebra.Com's Answer #845680 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Find the remainder when $\"40%5E13\"$ is divided by 81.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        It is clear that the mathematical meaning of this problem is not to follow literally \r
\n" ); document.write( "\n" ); document.write( "        the written formula.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Its meaning is to decrease / (to reduce) the degrees and values of participating numbers \r
\n" ); document.write( "\n" ); document.write( "        to make calculations easier using standard properties of operations of modular arithmetic.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Following this idea, I write  40 = 36 + 4,\r\n" );
document.write( "\r\n" );
document.write( "     = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next we should apply the Newtonian binomial formula.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It will give the sum of the terms  ,  k = 0, 1, 2, 3, . . . , 13.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "All the terms with k >= 2  will be zero by modulo 81, since 36 = 9*4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, we can exclude all these terms from our consideration.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the terms under our consideration are the terms with k= 0 and k= 1, or\r\n" );
document.write( "\r\n" );
document.write( "     +  =  + .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This expression is easy to calculate using a regular calculator or Excel spreadsheet; \r\n" );
document.write( "its value is 7918845952.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Finally,  7918845952 mod 81 is 22  (use long division or Excel function mod)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the  ANSWER  is 22.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Throwing pebbles into the water from a bridge, look at the circles
\n" ); document.write( "they form; otherwise your exercises will be empty deals.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );