Find the largest integer

Let $x$ be the smallest integer, then the other integers are $x+2,x+4,x+6,x+8$ and $x+10$ . From the given data, we have

$x+x+2+x+4+x+6+x+8+x+10=78$

$6x+30=78$

$6x=48$

$x=8$

Finally, the desired answer is $x+10=\boxed{18}$ .

Let the numbers are 2n-4,2n -2, 2n, 2n+2, 2n+4, 2n+6. The sum is 12n+6=78, n=6. So the largest integer is 18