Review question

# What do we get if we add the digits of the integers from 1 to 99? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7399

## Solution

The function $S(n)$ is defined for positive integers $n$ by $S(n) = \text{sum of the digits of }n.$ For example, $S(723) = 7+2+3=12$. The sum $S(1) + S(2) + S(3) + \dots + S(99)$ equals

1. $746$,

2. $862$,

3. $900$,

4. $924$.

### Approach 1

We can lay out the digits we have to add in a table and then add up each column.

 $0+0$ $0+1$ $0+2$ $0+3$ $\dots$ $0+9$ $1+0$ $1+1$ $1+2$ $1+3$ $\dots$ $1+9$ $2+0$ $2+1$ $2+2$ $2+3$ $\dots$ $2+9$ $\dots$ $\dots$ $\dots$ $\dots$ $\dots$ $\dots$ $9+0$ $9+1$ $9+2$ $9+3$ $\dots$ $9+9$ $1+2+\dots+9=45$ $45+10\times1$ $45+10\times2$ $45+10\times3$ $\dots$ $45+10\times9$

Adding up the column totals we have $45\times10 + 10\times1 + 10\times2 +\dots+10\times9 = 45\times10+10\times45 = 900$ so the answer is (c).

### Approach 2

Since zeroes do not contribute to the sum, we can rewrite this as $S(1) + S(2) + S(3) + \dots + S(99) = S(00) + S(01) + \dots + S(99).$

Then in the $100$ two-digit numbers $00, \dots, 99$, each digit $0,\dots ,9$ appears $20$ times ($10$ times as the first digit, $10$ times as the second).

So $S(1) + S(2) + S(3) + \dots + S(99) = 20 \times (0 + 1 + \dots + 9) = 20 \times 10 \times \frac{(0 + 9)}{2} = 900,$ and the answer is (c).