Mathematics - Functions Question with Solution | TestHub

From the given equation, we have

$\begin{array}{rlr}& \left(\frac{n}{2}-\left[\frac{n}{2}\right]\right)+\left(\frac{n}{3}-\left[\frac{n}{3}\right]\right)+\left(\frac{n}{5}-\left[\frac{n}{5}\right]\right)=0& \Rightarrow \left\{\frac{n}{2}\right\}+\left\{\frac{n}{3}\right\}+\left\{\frac{n}{5}\right\}=0\text{; where {.} is a F.P.F. }\end{array}$

But each of the fraction part functions is positive and their sum is zero. Hence each of the fraction part function is zero. Consequently, each of $\frac{n}{2},\frac{n}{3},\frac{n}{5}$ is an integer. The 1.c.m. of $2,3,5$ is 30 . Therefore we can take $\mathrm{n}=30\mathrm{k}$ where k is an integer.

Hence the number of solutions such that $1\le n\le 100$ is $=3$ (viz. $\mathrm{n}=30,60$ and 90 )