Numbers that are divisible So I am given the following question: For natural numbers less than or equal to 120, how many are divisible by 2, 3, or 5? I solved it by inclusion-exclusion principle and by using the least common multiple by having it as (2, 3, 5)=120 which is equal to 30. Are these the right way to solve them or is there other ways?

It is a right way. Inevitably there are others. For example, there are $\varphi(120)$ numbers in the interval $[1,120]$ which are relatively prime to $120$. Here $\varphi$ is the Euler $\varphi$-function. 
The numbers in our interval which are divisible by $2$, $3$, or $5$ are precisely the numbers in our interval which are not relatively prime to $120$.  
So $120-\varphi(120)$ gives the desired count. Compute $\varphi(120)$ by using the usual formula, and the fact that $120=2^3\cdot 3\cdot 5$. 
The Inclusion/Exclusion procedure is more versatile than the Euler $\phi$-function procedure. 
