GATE CS 1995 Q25b — Sets, Relations and Functions

We need to find the number of positive integers less than or equal to $720$ that are not divisible by $2$, $3$, or $5$.

Since $720 = 2^4 \times 3^2 \times 5^1$, its prime factors are exactly $2$, $3$, and $5$. Therefore, any integer $x \le 720$ is not divisible by $2$, $3$, or $5$ if and only if it is coprime to $720$.

This can be directly calculated using Euler's totient function $\phi(n)$:

Thus, there are 192 such integers.