JEE Mathematics 2023 Question with Solution

Given: We need the number of $3$-digit numbers divisible by either $3$ or $4$, but not divisible by $48$.

Find: The required count.

Total $3$-digit numbers:

Numbers divisible by $3$:

Numbers divisible by $4$:

Numbers divisible by both $3$ and $4$ are divisible by $12$:

Using inclusion-exclusion, numbers divisible by either $3$ or $4$:

Numbers divisible by $48$:

Therefore, numbers divisible by either $3$ or $4$ but not by $48$:

So, the correct option is B.

Given: Count $3$-digit numbers satisfying divisibility by $3$ or $4$, excluding those divisible by $48$.

Find: The exact number of such integers.

First count all $3$-digit integers from $100$ to $999$:

Now count multiples of each required divisor within these $900$ numbers as done in the solution:

Apply inclusion-exclusion:

Remove the numbers divisible by $48$:

Hence, the number of such $3$-digit numbers is $432$. Therefore, the correct option is B.