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. 6
• B. 9
• C. 14
• D. 18
• E. None of the remaining options is correct

Answer 1

The Correct Option is C

To solve this problem, we need to find the least possible value of a number \( x \) such that:

• The least common multiple (LCM) of \( x \) and 990 is 6930.
• The greatest common divisor (GCD) of \( x \) and 550 is 110.

We will use the relation between LCM and GCD: For any two numbers \( a \) and \( b \), the product of their LCM and GCD is equal to the product of the numbers themselves: \(LCM(a, b) \times GCD(a, b) = a \times b\).

Based on this, we can set up an equation for our problem:

\(LCM(x, 990) \times GCD(x, 990) = x \times 990\)

We also have \(LCM(x, 990) = 6930\)

Assuming \(GCD(x, 990) = d\), we get:

\(6930 \times d = x \times 990\)

From here, we find:

\(x = \frac{6930d}{990}\)

Let's calculate \(x\) by setting the conditions for the least value of \(x\). Since \(GCD(x, 550) = 110\), \(x\) must be a multiple of 110.

Step-by-Step Calculation:

1. Calculate \( x \) using the equations:
   • First simplify: \( \frac{6930}{990} = \frac{693}{99} = \frac{7}{1} \). (This simplifies using common factors)
   • So, \( x = 7 \times d \)
2. Since \( x \) must be a multiple of 110, let's find the possible values of \( d \) by using:
   • Possible values of \( d \) are divisors of 990 that also keep \( x \) as a multiple of 110.
   • Divisors of 990 include 1, 2, 3, 5, 6, 9, 10, 11, 15, ..., all reducing to numbers like 110.
   • Consider \( d = 110 \), then calculate:
   • \( x = 7 \times 110 = 770 \)
3. Now, verify the LCM condition for correctness:
   • \( LCM(770, 990) = 6930 \)
   • Upon verification, the calculation checks out since \( LCM(770, 990) \) runs through the breakdown of prime factors and results in 6930 indeed.
4. Finally, check sum of the digits of \( x = 770 \):
   • \((7 + 7 + 0) = 14\)

Conclusion: The least possible value of the number is 770, and the sum of its digits is 14. Thus, the correct answer is 14.

Answer 2

Step 1: Use the relationship between LCM and GCD. The relationship between LCM and GCD is:

LCM(a, b) ⋅ GCD(a, b) = a ⋅ b

Let the required number be x. Using the given data:

LCM(x, 990) ⋅ GCD(x, 550) = x ⋅ 990.

Substitute:

6,930 ⋅ 110 = x ⋅ 990.

Simplify:

\(x = \frac{6,930 \cdot 110}{990}\)

Step 2: Calculate x. Simplify the expression:

\(x = \frac{6,930 \cdot 110}{990} = \frac{693 \cdot 110}{99} = 770\)

Step 3: Find the sum of the digits of x.

Sum of digits of 770 = 14.

Answer: 14