What will be remainder when N=10^10+ 10^100 + 10^1000+ 10^10000..........+10^10000000000 is divided by 7

• 10^10+ 10^100 + 10^1000+ 10^10000..........+10^10000000000 % 7
  Means 10^10%7 + 10^100%7 + 10^1000%7 + ...
  
  10^1 % 7 = 3
  10^2 % 7 = 2
  10^3 % 7 = 6
  10^4 % 7 = 4
  10^5 % 7 = 5
  10^6 % 7 = 1
  10^7 % 7 = 3 [Again it goes on like 2,6,4,5,1,etc]
  
  So cycle is upto the power of 6... So R[N % 7] is 6k+n...
  10 => 6*1 + 4 ... R[10^10 % 7] = 4
  100 => 6*16+4 ... R[10^100 % 7] = 4
  1000 => 6*166+4 ... R[10^1000 % 7] = 4
  So For all the powers of 10^(10^n), 4 is the remainder...
  [4+4+4+4+4...+4]%7
  10*4 % 7
  40 % 7
  5 % 7
  
  Ans : 5
• 10 years agoHelpfull: Yes(29) No(1)
• Answer is damn right i.e. 5
  I used the following method:
  Acc to Fermat Little theorem Rem[M^(N-1)/N]=1 like Rem[10^6/7]=1
  So here we have Rem[10^10/7]=>Rem[10^6*10^4/7]=>Rem[10^4/7]=>Rem[3^4/7]=4
  Similarly other terms too give Remainder 4
  Total Remainder=4*10 times=40
  Rem[40/7]=5
  Hence 5 Answer.
  Bingo !!
• 10 years agoHelpfull: Yes(5) No(0)
• if you use a calculator 10^10 , 10^100 , 10^1000 and so on all leave a remainder of 4 when divided by 7.
  so the final remainder is the remainder of 4 * 10 by 7 = 5
• 7 years agoHelpfull: Yes(0) No(0)