Lots of digit sums

Number Theory Level 2

The digit sum of a positive integer is the sum of its digits. For example, the digit sum of 421 = 4 + 2 + 1 = 7 421=4+2+1=7 .

What is the total of the digit sums of all integers from 0 0 to 999 999 , inclusive?

10000 4500 14500 13500

Matthew Christopher by Matthew Christopher , 17, United Kingdom

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2 solutions

Each digit from 0 0 to 9 9 appears 100 100 times in each of the 3 3 columns.

The answer is therefore 3 ⋅ 100 ⋅ ∑ r = 0 9 r = 300 ⋅ 9 ( 9 + 1 ) 2 = 300 ⋅ 45 = 13500 3\cdot 100\cdot\sum_{r=0}^9r=300\cdot\dfrac{9(9+1)}{2}=300\cdot45=\color{#20A900}{\boxed{13500}}

4 Helpful 1 Interesting 0 Brilliant 1 Confused
Barry Leung
Aug 9, 2020

When we rewrite each number using three digits, like 37 = 037 37 = 037 , we see that each digit from 0 0 to 9 9 appears exactly 100 times in each column. For the calculation, check @Matthew Christopher 's solution.

1 Helpful 1 Interesting 0 Brilliant 0 Confused

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