JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Permutations and Combinations

Since $54=2\cdot 3^3$, the condition $\gcd (N,54)=2$ means $N$ is even but not a multiple of $3$. Among the $9000$ four-digit numbers, $\lfloor \frac{9000}{2}\rfloor =4500$ are even, and those that are also multiples of $3$ are the multiples of $6$, of which there are $\lfloor \frac{9000}{6}\rfloor =1500$. Removing these leaves $4500-1500=3000$ such numbers.