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

Four-digit numbers are $\left\{1000, 1001, . . . , 9999\right\}$

Total number of four digit numbers are $9000$.

Now,

$54 = 2 \times {3}^{3}$

So, for getting gcd with $2$, we must have numbers multiple of $2$ but not of $3$.

Multiple of $2$ are \left\{1000, 1002, . . . . . , 9998\right\}$</p><p>Let total numbers be$m$, then</p><p>$9998 = 1000 + \left(m - 1\right)2$</p><p>$⇒ m = 4500$</p><p>Total numbers are$4500$.</p><p>Multiple of$6$are \left\{1002, 1008 . . . . , 9996\right\}$

Let total numbers be $n$, then

$9996 = 1002 + \left(n - 1\right)6$

$⇒ n = 1500$

So, required number is $= 4500 - 1500 = 3000$