JEE Main 2025 Jan 24 Shift 1, Mathematics Q23: Permutations And Combinations

3-digit integers are 100 to 999.

"Divisible by 2 and 3" means divisible by $\text{lcm}(2,3)=6$. Count: $\lfloor \frac{999}{6}\rfloor -\lfloor \frac{99}{6}\rfloor =166-16=150$.

From these, exclude those also divisible by both 4 and 9, i.e. by $\text{lcm}(4,9)=36$ (any multiple of 36 is automatically a multiple of 6). Count: $\lfloor \frac{999}{36}\rfloor -\lfloor \frac{99}{36}\rfloor =27-2=25$.

Required count $=150-25=125$.

$\therefore$ Answer: $125$.