If the number 123k578 is divisible by 11, then what is the value of k?

Sum of odd place digits = $8 + 5 + 3 + 1 = 17$

Sum of even place digits = $7 + k + 2 = 9 + k$

Hence Sum of odd place digits - Sum of even place digits = $0, 11, 22,....$

Hence $17 - (9 + k) = 0, 11, 22,....$

$8 - k = 0, 11, 22,....$

Since k is a digit (0-9), the possible values for $8 - k$ are between $8-9=-1$ and $8-0=8$.

The only multiple of 11 in this range is 0.

Hence $8 - k = 0$

Possible value of $k = 8$

And this is the greatest as well as least possible value of k

Hence the difference between the greatest value and smallest possible values = $8 - 8 = 0$