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One day, Mark writes down the numbers \(1, 2, \dots,999.\) What is the sum of all the digits that Mark wrote down?

 Jun 5, 2020
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We can go by the digit place.

 

In the one's digit, we have that each digit is used 100 times. This is because the "two-digit number" in the hundreds and tens place goes from 00 to 99. For example, for the digit 1, we have \(1,11,111, \cdots, 981,991\). This is the same for all digits in the one's place.

 

Doing something similar in the ten's place, we see that there are still 100. The first digit can go from 0 to 9, and so can the last.

 

The same thing happens when the digit is in the hundred's place.

 

We didn't miss any or overcount (for example 000), since 0 doesn't add to the answer and thus doesn't matter.

 

Thus, the answer is \((1+2+3+\cdots + 9) \cdot (300) = \boxed{13500}\).

 Jun 5, 2020

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