JEE Mathematics 2023 Question with Solution

Given: We need the number of $3$-digit numbers divisible by either $2$ or $3$, but not divisible by $7$.

Find: The required count.

Using the inclusion-exclusion counts given in the solution:

$$\text{Divisible by } 2 = 450$$ $$\text{Divisible by } 3 = 300$$ $$\text{Divisible by } 7 = 128$$ $$\text{Divisible by } 2 \text{ and } 7 = 64$$ $$\text{Divisible by } 3 \text{ and } 7 = 43$$ $$\text{Divisible by } 2 \text{ and } 3 = 150$$ $$\text{Divisible by } 2, 3 \text{ and } 7 = 21$$

Applying inclusion-exclusion for numbers divisible by $2$ or $3$ but not by $7$:

$$450 + 300 - 150 - 64 - 43 + 21 = 514$$

Therefore, the required number of such $3$-digit numbers is $514$.

Given: Counts of $3$-digit numbers divisible by $2$, $3$, $7$, and their intersections are provided.

Find: Numbers divisible by either $2$ or $3$ but not by $7$.

First count numbers divisible by $2$ or $3$:

$$450 + 300 - 150$$

Now remove those also divisible by $7$. These are numbers divisible by $2$ and $7$, or by $3$ and $7$. Since numbers divisible by $2, 3,$ and $7$ get removed twice, add them back once:

$$(450 + 300 - 150) - 64 - 43 + 21$$ $$= 514$$

Hence, the answer is $514$.