Basic counting

The number of integers between 1 and 100 (inclusive) divisible by 9 or 4 $=$ the number of integers between 1 and 100 (inclusive) divisible by 9 $+$ the number of integers between 1 and 100 divisible (inclusive) by 4 $-$ the number of integers between 1 to 100 (inclusive) divisible by both 4 and 9.

In formula:

$\begin{aligned} n & = \left \lfloor \frac {100}9\right \rfloor + \left \lfloor \frac {100}4\right \rfloor - \left \lfloor \frac {100}{36} \right \rfloor = 11 + 25 - 2 = \boxed{34} \end{aligned}$

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

First let us note that:

1. Between $"1 - 100"$ , $11$ numbers are divisible by $9$

2. Between $"1 - 100"$ , $25$ numbers are divisible by $4$

You might assume that the numbers which are divisble by $9$ or $4$ is $36$ $\rightarrow 25 + 11$ .

However, we're counting every no. that is divisible by $4$ and $9$ twice, ex: $36$ . Hence we have to find out how many such no.s are there.

There are two no.s, $36$ and $72$ .

$\therefore$ the numbers between $1 - 100$ which are divisible by $9$ or $4$ is equal to:

$36 - 2 = \large \color{#20A900} \boxed{34}$