1. We need the number of 4-digit numbers $n$ such that $\gcd(n,54)=2.$

2. First factorize $54$: $54=2\cdot 3^3.$

For $\gcd(n,54)=2,$ we need:

• $n$ to be divisible by $2$,
• $n$ to not be divisible by $3$.

Why? Since the gcd is exactly $2$, the number can share the factor $2$ with $54$, but must share no factor $3$.

So we need to count 4-digit even numbers that are not divisible by 3.

3. Total 4-digit even numbers:

The 4-digit numbers run from $1000$ to $9999$. The even ones run from $1000$ to $9998$ with common difference $2$.

Number of such terms is $\frac{9998-1000}{2}+1=\frac{8998}{2}+1=4499+1=4500.$

So total 4-digit even numbers $=4500$.

4. Now count 4-digit numbers divisible by both $2$ and $3$, i.e. divisible by $6$.

Multiples of $6$ from $1000$ to $9999$:

• First multiple of $6$ greater than or equal to $1000$ is $1002$.
• Last multiple of $6$ less than or equal to $9999$ is $9996$.

Count: $\frac{9996-1002}{6}+1=\frac{8994}{6}+1=1499+1=1500.$

So 4-digit even numbers divisible by $3$ are $1500$.

5. Therefore required count is $4500-1500=3000.$

Thus, the total number of 4-digit numbers satisfying $\gcd(n,54)=2$ is $\boxed{3000}.$

6. Comparison with stored answer:

Stored correct answer = $3000$. Our derived answer = $3000$. So they agree.