The ten digit number 2x600000y8 is exactly divisible by 24. If x ≠ 0 and y ≠ 0, then the least value of (x + y) is equal to:

If a number is divisible by both 8 and 3 then the number is divisible by 24 also.

So divisibility rule for 8 is last three digit should be divisible by 8

So, 536 and 576 are the last three digits divisible by 8, so we have to take least value hence, x = 3

Now, divisiblity rule for 3 is sum of the digits should be divisible by 3

So, $$2+x+6+0+0+0+0+0+y+8=16+x+y+6$$ should be divisible by 3

$$16 + 3 +y=19+y$$ should be divisible by 3

Now, least value for y which satisfied the condition is 2

So, $$x + y = 3 + 2 = 5$$