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kingbucky
­Consider the sum \(7+ 77+ 777 +7777+ 77777+ ....+ 77,777,777,777,777,777,777\), where the last addend has 20 digits. Find the digit in the hundreds place of the resulting sum.

A) 0
B) 8
C) 6
D) 4
E) 1­

[Source: MATHCOUNTS]
Consider how you will add 20 of these numbers:

7777....................777
0777....................777
0077....................777
0007....................777
...
...
0000....................077
0000....................007
____________________

You will start by adding all the units digits. There are 20 7s in units place. Their sum will be 20*7=140.
So you will write a 0 in the units place and carry 14 to the tens place.


7777....................777
0777....................777
0077....................777
0007....................777
...
...
0000....................077
0000....................007
____________________
..............................0


Next we will add 19 7s of the tens place and the 14 carry to get 19*7 + 14 =147
Now we will write 7 in the units place and carry 14.

7777....................777
0777....................777
0077....................777
0007....................777
...
...
0000....................077
0000....................007
____________________
.............................70

Next we will add 18 7s of the hundreds place and the 14 carry to get 18*7 + 14 =140
So we will write 0 in the hundreds place and carry 14.

7777....................777
0777....................777
0077....................777
0007....................777
...
...
0000....................077
0000....................007
____________________
............................070


So the hundreds digit is 0.
Answer (A)
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