Animated Solution for Mathematics - Sequence and Series: The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______.

We are tasked with finding the number of integers $n$ in the range $[100,999]$ such that $\text{GCD}(n,36)=2$.

The number $36$ can be expressed through its prime factorization as:

For $\text{GCD}(n,36)=2$ to hold, $n$ must satisfy specific divisibility constraints derived from the prime factors of $36$.

The condition $\text{GCD}(n,36)=2$ imposes three strict requirements on $n$:

1. $n$ must be a multiple of $2$.

2. $n$ must NOT be a multiple of $4$ (since $2^2=4$ is a factor of $36$).

3. $n$ must NOT be a multiple of $3$ (since $3$ is a prime factor of $36$).

First, we identify all multiples of $2$ in the range $[100,999]$. The sequence is $100,102,\dots ,998$.

The number of such terms is:

Next, we remove the multiples of $4$ from this set. The multiples of $4$ in the range $[100,999]$ are $100,104,\dots ,996$.

The number of such terms is:

Subtracting these, we find the number of integers divisible by $2$ but not by $4$:

Finally, we must remove numbers from our set of $225$ candidates that are also divisible by $3$. Since our candidates are already multiples of $2$, removing those divisible by $3$ is equivalent to removing multiples of $6$.

However, we must only remove multiples of $6$ that are not multiples of $12$ (because multiples of $12$ are already excluded as they are multiples of $4$).

The sequence of numbers to exclude starts at $102$ ($6\times 17$) and ends at $990$ ($6\times 165$). The common difference is $12$.

We calculate the number of terms $k$ using the arithmetic progression formula:

We subtract the $75$ numbers that failed the third filter from our previous count of $225$:

There are exactly $150$ such numbers in the range $[100,999]$.