Question
A natural number ' $x$ ' is chosen at random from the first 1000 natural numbers. If $\lbrack \cdot \rbrack$ denotes the greatest integer function, then the probability that $\left\lbrack \frac{x}{2} \right\rbrack + \left\lbrack \frac{x}{3} \right\rbrack + \left\lbrack \frac{x}{5} \right\rbrack = \frac{31x}{30}$ is
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Solution
Sol. [Key Concept: Pick $x$ from the first $N$ naturals satisfying the divisibility condition.]
$$\frac{x}{2} - \left\{ \frac{x}{2} \right\} + \frac{x}{3} - \left\{ \frac{x}{3} \right\} + \frac{x}{5} - \left\{ \frac{x}{5} \right\} = \frac{31x}{30}$$
$$\begin{matrix} \Rightarrow & & & \begin{Bmatrix} x \\ 2 \end{Bmatrix} + \begin{Bmatrix} x \\ 3 \end{Bmatrix} + \begin{Bmatrix} x \\ 5 \end{Bmatrix} = 0 \\ \Rightarrow & \frac{x}{2} \in I,\frac{x}{3} \in I\&\frac{x}{5} \in I & & \\ \Rightarrow & & & x = 30k,k \in I \\ \therefore & & & P = \frac{33}{1000} \end{matrix}$$
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