Question

A natural number $x$ is chosen at random from the first $1000$ natural numbers. If $[.]$ denotes the greatest integer function, then the probability that $[\frac{x}{2}]+[\frac{x}{3}]+[\frac{x}{5}]=\frac{31x}{30}$ is

• A. $33/1000$
• B. $34/1000$
• C. $32/1000$
• D. $31/1000$

Answer 1

Answer: (A)

$33/1000$

We observe that $\frac{x}{2}+\frac{x}{3}+\frac{x}{5}=\frac{31x}{30}$
This implies that x should be a number which is a multiple of $2,3,5$ i.e. a multiple of $30$.
From the first $1000$ numbers, the number of multiples of $30$ is $\left[\frac{1000}{30}\right]=33$.
Hence, the required probability = $\frac{33}{1000}$