How many six-digit numbers are there in which no digit is repeated, even digits appear at even places,odd digits appear at odd places and the number is divisible by $4$ ?

We need to fill the above spaces in such a way that places $2,4,6$ have even numbers and the rest have odd. Also, none of the numbers must be repeated. Since the number is divisible by $4$, the possible numbers for $5$ and $6$ are $6$ can be either $2,6$ and $5$ can be an odd number.

$\Rightarrow$ Combinations $={}^2{C}_1\times {}^5{C}_1\times {}^4{C}_1\times {}^4{C}_1\times {}^3{C}_1\times {}^3{C}_1$ 

                                      $6$        $5$        $4$        $3$        $2$       $1$

                              $=2\times 5\times 16\times 9$

$\Rightarrow$               Total  $=1440$

Hence, the answer is $1440.$