document.write( "Question 319471: Determine the last digit of 3^3^3 \n" ); document.write( "
Algebra.Com's Answer #228772 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Does \"3^3^3\" or \"drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E3%5E3%29+%29\" mean \"%283%5E3%29%5E3=19683\" or \"drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29\"?\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "I will assume it's \"drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29\"; otherwise we'd just do it with the calculator\r\n" );
document.write( "\"%283%5E3%29%5E3=19683\" and the answer would obviously be 3.\r\n" );
document.write( "\r\n" );
document.write( "So I assume you mean:\r\n" );
document.write( "\r\n" );
document.write( "\"drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29\"\r\n" );
document.write( "\r\n" );
document.write( "30 = 1\r\n" );
document.write( "31 = 3\r\n" );
document.write( "32 = 9\r\n" );
document.write( "33 = 27\r\n" );
document.write( "\r\n" );
document.write( "34 = 81\r\n" );
document.write( "35 = 243\r\n" );
document.write( "36 = 729\r\n" );
document.write( "37 = 2187\r\n" );
document.write( "\r\n" );
document.write( "We can see that the last digits repeat in cycles of 4:  1,3,5,7,1,3,5,7,...\r\n" );
document.write( "So the last digit of \"3%5En\" is the same as the last digit of n mod 4.\r\n" );
document.write( "\r\n" );
document.write( "Therefore since \"drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29\" is the same as \"3%5E27\", and since 27 mod 4 = 3, \r\n" );
document.write( "the last digit of it is the same as the last digit of \"3%5E3=27\", so the answer is 7. \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );