document.write( "Question 242868: Find the units digit of the number N given that 57^100 + 56^100 + 55^100 = N \n" ); document.write( "
Algebra.Com's Answer #177851 by Edwin McCravy(20060)![]() ![]() You can put this solution on YOUR website! \r\n" ); document.write( "57^100 + 56^100 + 55^100 = N\r\n" ); document.write( "\r\n" ); document.write( "All positive integer powers of positive integers ending in 6 end in 6.\r\n" ); document.write( "Therefore 56^100 ends in 6.\r\n" ); document.write( "\r\n" ); document.write( "All positive integer powers of positive integers ending in 5 end in 5.\r\n" ); document.write( "Therefore 55^100 ends in 5.\r\n" ); document.write( "\r\n" ); document.write( "So the only difficulty here is with positive integer powers of positive\r\n" ); document.write( "integers ending in 7. So we write down a few:\r\n" ); document.write( "\r\n" ); document.write( "7^1 = 7\r\n" ); document.write( "7^2 = 49\r\n" ); document.write( "7^3 = 343\r\n" ); document.write( "7^4 = 2401\r\n" ); document.write( "7^5 must end in 7, and the units digits must start over.\r\n" ); document.write( "\r\n" ); document.write( "So the units digits of powers of positive integers ending \r\n" ); document.write( "in 7 go 7,9,3,1,7,9,3,1,...\r\n" ); document.write( "\r\n" ); document.write( "which means every 4th one will be a 1.\r\n" ); document.write( "\r\n" ); document.write( "Since the 100th one is a 4th one, 57^100 will end in a 1.\r\n" ); document.write( "\r\n" ); document.write( "So now we know that: \r\n" ); document.write( "\r\n" ); document.write( "57^100 ends in a 1, \r\n" ); document.write( "56^100 ends in a 6, and\r\n" ); document.write( "55^100 ends in a 5,\r\n" ); document.write( "\r\n" ); document.write( "Therefore we know that when we add them we will begin by adding their\r\n" ); document.write( "last digits 1+6+5 = 12, and we will put down the 2 and carry the 1.\r\n" ); document.write( "So the last digit of 57^100 + 56^100 + 55^100 = N must be 2.\r\n" ); document.write( "\r\n" ); document.write( "Edwin\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |