JEE Mathematics 2024 Question with Solution

Given: An integer is chosen uniformly from $1$ to $50$.

Find: The probability that it is divisible by at least one of $4$, $6$, $7$.

Use the principle of inclusion-exclusion.

Number of multiples of $4$ in $1$ to $50$ is $12$.

Number of multiples of $6$ in $1$ to $50$ is $8$.

Number of multiples of $7$ in $1$ to $50$ is $7$.

Now count common multiples of pairs:

• LCM$(4,6) = 12$, so common multiples of $4$ and $6$ are $4$.
• LCM$(6,7) = 42$, so common multiples of $6$ and $7$ are $1$.
• LCM$(4,7) = 28$, so common multiples of $4$ and $7$ are $1$.

For all three numbers,

Since $84 > 50$, the number of multiples common to all three is $0$.

Applying inclusion-exclusion,

Let $A$ be the event that the chosen integer is a multiple of $4$, $B$ the event that it is a multiple of $6$, and $C$ the event that it is a multiple of $7$.

Then,