Animated Solution for Mathematics - Sequence and Series: The number of 3 -digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is

Our objective is to identify three-digit numbers in the range $[100,999]$ that are divisible by both $2$ and $3$, while excluding those that are divisible by both $4$ and $9$.

Since a number divisible by two coprime integers must be divisible by their product, we define our sets based on the Least Common Multiple (LCM). For the first condition, the LCM of $2$ and $3$ is $6$. For the exclusion condition, the LCM of $4$ and $9$ is $36$.

Let $A$ be the set of three-digit numbers divisible by $6$. The smallest multiple of $6$ in the range $[100,999]$ is $102$, and the largest is $996$.

Let $B$ be the set of three-digit numbers divisible by $36$. The smallest multiple of $36$ in the range $[100,999]$ is $108$, and the largest is $972$.

The sequence of numbers in $A$ forms an Arithmetic Progression where the first term $a=102$, the common difference $d=6$, and the last term $a_n=996$. We use the formula:

Substituting our values:

Thus, there are $150$ numbers divisible by $6$.

Similarly, for set $B$, the first term $a=108$, the common difference $d=36$, and the last term $a_n=972$. Applying the same formula:

There are $25$ numbers divisible by $36$.

Since every multiple of $36$ is inherently a multiple of $6$, set $B$ is a subset of set $A$. To find the number of elements in $A$ that are not in $B$, we perform the subtraction:

The total count of numbers satisfying the given conditions is $125$.