Mathematics - Permutation & Combination Question with Solution | TestHub

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 $\left\{1000,1002,.....,9998\right\}$

Let total numbers be $m$, then

$9998=1000+\left(m-1\right)2$

$\Rightarrow m=4500$

Total numbers are $4500$.

Multiple of $6$ are $\left\{1002,1008....,9996\right\}$

Let total numbers be $n$, then

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

$\Rightarrow n=1500$

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