The total number of 4 -digit numbers whose greatest common divisor...

Let $N$ be the four digit number $\gcd \left(N, 18\right) = 3$

Hence $N$ is an odd integer which is divisible by $3$ but not by $9$.

$4$ digit odd multiples of $3$

$1005, 1011, \ldots , 9999 \to 1500$

digit odd multiples of $9$

$1017, 1035, \ldots , 9999 \to 500$

Hence number of such $N = 1500 - 500 = 1000$