Question

Total number of 3-digit numbers, whose g.c.d with 36 is 2, is

• A. 140
• B. 150
• C. 165
• D. 170

Hint:

Use the inclusion-exclusion principle or Euler's totient idea to count numbers coprime to a set of factors.

Answer 1

The Correct Option is B

Step 1: Concept
For $\text{gcd}(n, 36) = 2$, $n$ must be even, but not divisible by 4, and not divisible by 3.

Step 2: Meaning
$36 = 2^2 \times 3^2$. If $\text{gcd}(n, 36) = 2$, then $2^1$ is the highest power of 2 in $n$, and 3 is not a factor of $n$.

Step 3: Analysis
Let $n = 2k$ where $\text{gcd}(k, 18) = 1$. This means $k$ is not divisible by 2 or 3. $n \in [100, 999] \implies k \in [50, 499]$. Total values for $k$ is 450. Number of $k$ not divisible by 2 or 3 is $450(1 - 1/2)(1 - 1/3) = 450(1/2)(2/3) = 150$.

Step 4: Conclusion
There are 150 such 3-digit numbers. Final Answer: (B)