Mathematics - Functions Question with Solution | TestHub

From the given equation, we have

$$\left(\frac{n}{2}-\left[\frac{n}{2}\right]\right)+\left(\frac{n}{3}-\left[\frac{n}{3}\right]\right)+\left(\frac{n}{5}-\left[\frac{n}{5}\right]\right)=0$$

This implies

$$\left\{\frac{n}{2}\right\}+\left\{\frac{n}{3}\right\}+\left\{\frac{n}{5}\right\}=0$$

where $\{.\}$ is a fractional part function.

But each of the fractional part functions is positive, and their sum is zero. Hence, each of the fractional part functions must be zero. Consequently, each of $\frac{n}{2}$, $\frac{n}{3}$, $\frac{n}{5}$ is an integer. The least common multiple (LCM) of $2,3,5$ is $30$. Therefore, we can take $n=30k$, where $k$ is an integer. Hence, the number of solutions such that $1\le n\le 100$ is $3$ (viz. $n=30,60,$ and $90$).