The number of 3-digit natural numbers that are divisible by both 2 and 3 but not by both 4 and 9 is
Step-by-Step Solution
Key Concept: Count 3-digit multiples of 6, then subtract those divisible by $\text{lcm}(4,9)=36$ (the only numbers divisible by both 4 and 9).
Divisible by 6: from 102 to 996. Count $=\left\lfloor\dfrac{996}{6}\right\rfloor-\left\lfloor\dfrac{99}{6}\right\rfloor=166-16=150$.
Divisible by both 4 and 9 $\Leftrightarrow$ divisible by 36: from 108 to 972. Count $=\left\lfloor\dfrac{972}{36}\right\rfloor-\left\lfloor\dfrac{99}{36}\right\rfloor=27-2=25$.
Answer: $150-25=125$.
Correct Answer: 125