Answer: D — 7

For the 7-digit number $4x5y790$, label digit positions from the right: position $1=0$, $2=9$, $3=7$, $4=y$, $5=5$, $6=x$, $7=4$.

Apply the divisibility rule for $11$.

(Sum of digits at odd positions) $-$ (sum of digits at even positions) must be divisible by $11$.

• Odd-position sum: $0 + 7 + 5 + 4 = 16$
• Even-position sum: $9 + y + x = x + y + 9$

So we need $16 - (x + y + 9) = 7 - (x+y)$ to be a multiple of $11$.

Solve.

$x, y$ are digits, so $0 \le x+y \le 18$, giving $7 - (x+y) \in [-11, 7]$. Multiples of $11$ in this range: $0$ and $-11$.

• $7 - (x+y) = 0 \Rightarrow x+y = 7$
• $7 - (x+y) = -11 \Rightarrow x+y = 18$

In both cases, $(x+y) \bmod 11 = 7$.