LCM and GCD Number Properties

Let the unknown number be $x$. $990 = 2 \times 3^2 \times 5 \times 11$ $6930 = 2 \times 3^2 \times 5 \times 7 \times 11$ Since $\text{LCM}(x, 990) = 6930$, $x$ must contain $7^1$ as a factor. $550 = 2 \times 5^2 \times 11$ $\text{GCD}(x, 550) = 110 = 2 \times 5 \times 11$. So $x$ must be divisible by $2, 5, 11$, and $7$, giving $x_{\text{min}} = 2 \times 5 \times 11 \times 7 = 770$. Sum of digits of $770 = 7 + 7 + 0 = 14$.