JEE Mathematics 2024 Question with Solution

Given: $A$ = {$n \in [100, 700] \cap N$ : $n$ is neither a multiple of $3$ nor a multiple of $4$}.

Find: The number of elements in $A$.

First find the total number of natural numbers in $[100,700]$:

Now count the multiples of $3$ in $[100,700]$. They are

This is an AP with

Using

we get

Thus, the number of multiples of $3$ is $200$.

Now count the multiples of $4$ in $[100,700]$. They are

This is an AP with

Using

we get

Thus, the number of multiples of $4$ is $151$.

Now count the numbers which are multiples of both $3$ and $4$, that is, multiples of $12$. They are

This is an AP with

Using

we get

Thus, the number of multiples of $12$ is $50$.

Apply the inclusion-exclusion principle:

Substituting the values,

Therefore,

Hence, the number of elements in $A$ is $300$. The correct option is A.