Question

The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110. What is the sum of the digits of the least possible value of that number ?

• A. 16
• B. 14
• C. 3
• D. 9
• E. 18

Hint:

Always use prime factorization for LCM and GCD problems. A number's prime factors must satisfy the maximum powers found in the LCM and the minimum powers found in the GCD.

Answer 1

The Correct Option is B

Step 1: Prime factorize the given numbers.
$990 = 2 \times 3^2 \times 5 \times 11$
$LCM = 6930 = 2 \times 3^2 \times 5 \times 7 \times 11$
$550 = 2 \times 5^2 \times 11$
$GCD = 110 = 2 \times 5 \times 11$

Step 2: Determine the constraints on the unknown number $N$.
Since the LCM of $N$ and 990 contains the prime factor 7 (which is not in 990), $N$ must be a multiple of 7.
Since the GCD of $N$ and 550 is 110, $N$ must be a multiple of 110, but it cannot contain $5^2$ (otherwise the GCD would include $5^2$).
Therefore, $N$ must be a multiple of $110 \times 7 = 770$.

Step 3: Find the least possible value and sum its digits.
Let's test $N = 770$.
LCM(770, 990) = LCM($110 \times 7$, $110 \times 9$) = $110 \times 63 = 6930$. (Matches)
GCD(770, 550) = 110. (Matches)
The least possible value of the number is 770.
Sum of digits = $7 + 7 + 0 = 14$.