Question

How many $3$-digit natural numbers are completely divisible by $4$?

Answer 1

Step 1: Condition for divisibility by $4$

A number is said to be divisible by $4$ when its last two digits are divisible by $4$.

For example, let us consider a random number $428$.

The last two digits are $28$.

We know that $4\times 7=28$, hence $28$ is divisible by $4$.

Therefore $428$ is divisible by $4$.

Step 2: Estimate the count of natural numbers

Let us enlist such $3$-digit natural:

$a=100,d=4,a_n=996$

$a_n=a+\left(n-1\right)d$

$\Rightarrow$$996=100+\left(n-1\right)4$

$\Rightarrow$$896=4n-4$

$$\begin{gathered} \Rightarrow 896+4=4n \\ \Rightarrow 900=4n \end{gathered}$$

$\Rightarrow$ $n=\frac{900}{4}=225$

There are a total of $225$ digits in this range.

Hence, the number of $3$-digit natural numbers that are completely divisible by $4$ is $225$.