Step 1: Utilize the LCM-GCD relationship. The relationship is stated as:
LCM(<i>a</i>, <i>b</i>) ⋅ GCD(<i>a</i>, <i>b</i>) = <i>a</i> ⋅ <i>b</i>
Let the unknown number be <i>x</i>. Based on the provided information:
LCM(<i>x</i>, 990) ⋅ GCD(<i>x</i>, 550) = <i>x</i> ⋅ 990.
Substitution yields:
6,930 ⋅ 110 = <i>x</i> ⋅ 990.
Simplification results in:
\(x = \frac{6,930 \cdot 110}{990}\)
Step 2: Compute <i>x</i>. Simplify the expression:
\(x = \frac{6,930 \cdot 110}{990} = \frac{693 \cdot 11}{99} = 77 \cdot 11 = 847.\)
Step 3: Determine the sum of the digits of <i>x</i>.
The sum of the digits of 847 is 8 + 4 + 7 = 19.
Answer: 19