Question

How many $3$-digit natural numbers are completely divisible by $3$?

Answer 1

Step 1: Condition for divisibility by $3$

A number is said to be divisible by $3$ when sum of its digits is divisible by $3$.

For example, let us consider a random number $423$.

The sum of the digits $=4+2+3=9$.

We know that $3\times 3=9$, hence $9$ is divisible by $3$.

Therefore $423$ is divisible by $3$.

Step 2: Estimate the count of natural numbers

Let us enlist such $3$-digit natural:

$102,105,108,111,114......987,990,993,996,999$

The first three-digit number which is divisible by $3$ is $102$.

The last three-digit number which is divisible by $3$ is $999$

The number of three-digit numbers divisible by $3$

$$\begin{gathered} =\frac{999-102}{3}+1 \\ =\frac{897}{3}+1 \\ =299+1 \\ =300 \end{gathered}$$

There are a total of $300$ digits in this range.

Hence, the number of $3$-digit natural numbers that are completely divisible by $3$ is $300$.