Apply Divisibility Rule of 11

A number is divisible by $11$ if the difference between the sum of its digits at odd places (from the right) and the sum of its digits at even places (from the right) is either $0$ or a multiple of $11$.

$\text{Sum}_{\text{odd}} - \text{Sum}_{\text{even}} = 11m, \quad m \in \mathbb{Z}$

Calculate Sum of Digits

For the number $58K764$, identifying digits from right to left:

• Odd place digits ($1\text{st}, 3\text{rd}, 5\text{th}$): $4, 7, 8$
• Even place digits ($2\text{nd}, 4\text{th}, 6\text{th}$): $6, K, 5$

$\text{Sum}_{\text{odd}} = 4 + 7 + 8 = 19\text{Sum}_{\text{even}} = 6 + K + 5 = 11 + K$

Solve for K

Subtract the sum of even place digits from the sum of odd place digits and set the result to $0$ (since $K$ is a single digit, $0 \le K \le 9$):

$19 - (11 + K) = 0 \implies 8 - K = 0 \implies K = 8$

Calculate required expression

Substitute $K = 8$ into the required expression $K^2 - 3K$:

$K^2 - 3K = 8^2 - 3(8) = 64 - 24 = 40$