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 $S(1)+S(2)+S(3)+\ldots+S(99)$ is

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

Answer 1

Answer: (D)

$S(1)+S(2)+S(3)+\ldots+S(99)=$ 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 $3, \ldots, 20$ times 9 and 9 times zero.
So, $S(1)+S(2)+S(3)+\ldots+S(99)$
$=20 \times 1+20 \times 2+20 \times 3+\ldots+20 \times 9+9 \times 0$
$=20[1+2+3+\ldots+9]$
$=\frac{20 \times 9 \times(9+1)}{2}\left[\because \sum_{n=1}^{n} n=\frac{n(n+1)}{2}\right]$
$=\frac{20 \times 9 \times 10}{2}=9 \times 10 \times 10=900$