A, B, C, D, E and F are six consecutive even numbers. If the sum of A and F is $1586$, then what is the sum of all the numbers?

Let the six consecutive even numbers be represented as:
$$A=x,B=x+2,C=x+4,D=x+6,E=x+8,F=x+10$$

Given that the sum of $A$ and $F$ is $1586$:

$$x+(x+10)=1586$$

$$2x+10=1586⟹2x=1576⟹x=788$$

The total sum of all six numbers is:

$$\text{Sum}=A+B+C+D+E+F$$

$$\text{Sum}=x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10)=6x+30$$

We can express $6x+30$ in terms of our known quantity $(2x+10)$:

$$\text{Sum}=3(2x+10)=3\times 1586=4758$$