+81 Consider the sum \(7+77+777+7777+\cdots+77777777777777777777\), where the last addend has 20 digits. Find the digit in the hundreds place of the resulting sum. Give a convincing argument with your answer.
The hundreds place of the sum will be 7.
To see this, we can group the terms of the sum together by their hundreds place:
7 + 77 + 777 + 7777 = 8500
7777 + 77777 + 777777 = 850000
777777 + 7777777 + 77777777 = 85000000
...
The sum of each group is 85000000, which has a 7 in the hundreds place. Since there are 20 groups, the hundreds place of the final sum will also be 7.
+81 Ok, so right now you've got me confused. Do you mean that \(7+77+777+7777 = 8500?\) I believe the sum of those four numbers is 8638... Also, what do you mean that "The sum of each group is 85000000, which has a 7 in the hundreds place"?
1 - In the first addition, you have 20 "7s". Therefore: 20 * 7 =140. So, the ones digit is "0" and 14 is the remainder.
2 - In the 2nd addition, we have 19 "7s" left. Therefore: 19 * 7 + 14 [last remainder] =147. So, the tens digit is "7", with the remainder of 14.
3 - In the 3rd addition, we have 18 "7s" left. Therefore: 18 * 7 + 14 [last remainder] =140. So, the hundreds' digit must be a "0".
4 - So, the last 3 digits of the entire addition should be: 070
+81 Thank you for telling me how this works, but could you please explain why you multiplied 20 by a 4 and got 140?
+81 Ok, I get that, but how did you get 14 as a remainder? I didn't see any division... (sorry, but I really don't understand)
+81 Ohh, ok I understand. The 14 each time carries onto the next digit, which causes you to add 14. Thank you so much for the help, and your patience! 
