JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Permutations and Combinations

Since $36=2^2\times 3^2$, for $\gcd (n,36)=2$, the number $n$ must be an even number (divisible by 2 but not by 4) and must not be divisible by 3.

Alternatively, $n$ must be an odd multiple of 2 (i.e., $n=2k$ where $k$ is odd) and not divisible by 3.

Odd multiples of 2 in the 3-digit range: $102,106,110,114,\dots ,998$. This is an AP with first term 102, common difference 4, last term 998. Number of terms = $\frac{998-102}{4}+1=225$.

Among these 225 numbers, we exclude those divisible by 3. The odd multiples of 2 that are also divisible by 3 are multiples of 6 of the form $6(2m+1)$, i.e., $102,114,126,\dots ,990$. Number of such terms = $\frac{990-102}{12}+1=75$.

Required count = $225-75=150$.