Question

Let $S(n)$ denote the sum of the digits of a positive integer n. e.g.
$S(178)=1+$ $7+8=16.$ Then, the value of $S(1)+S(2)+S(3)+\dots +S(99)$ is

• A. 476
• B. 998
• C. 782
• D. 900
• E. 855

Answer 1

Answer: (D)

$S(1)+S(2)+S(3)+\dots +S(99)=$ Sum of digits of counting numbers 1 to 99 .
We know that, in 1 to 99 , there is 20 times 1,20 times 2, 20 times $3,\dots ,20$ times 9 and 9 times zero.
So, $S(1)+S(2)+S(3)+\dots +S(99)$
$=20\times 1+20\times 2+20\times 3+\dots +20\times 9+9\times 0$
$=20[1+2+3+\dots +9]$
$=\frac{20\times 9\times (9+1)}{2}\left[\because {\sum }_{n=1}^{n}n=\frac{n(n+1)}{2}\right]$
$=\frac{20\times 9\times 10}{2}=9\times 10\times 10=900$