If the average of $6$ consecutive even numbers is $183$, find the smallest number.

Let the $6$ consecutive even numbers be $x,x+2,x+4,x+6,x+8,$ and $x+10$, where $x$ is the smallest even number.
The average of these $6$ numbers is given by:

$$\text{Average}=\frac{x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10)}{6}$$

$$\text{Average}=\frac{6x+30}{6}=x+5$$

Given that the average is $183$:

$$x+5=183$$

$$x=183-5=178$$

Therefore, the smallest number is $178$.