The LCM of two numbers is 8919, and their sum is 1000. The two numbers are:

Options

A. 993 and 7
B. 989 and 11
C. 991 and 9
D. 987 and 13

Answer

To find the two numbers whose LCM is 8919 and whose sum is 1000, we can follow these steps:

### Step 1: Set up the equations
Let the two numbers be \( X \) and \( Y \). According to the problem, we have:
1. \( X + Y = 1000 \)
2. \( \text{LCM}(X, Y) = 8919 \)

From the first equation, we can express \( Y \) in terms of \( X \):
\[ Y = 1000 - X \]

### Step 2: Use the LCM formula
The LCM of two numbers can also be expressed in terms of their product and GCD (Greatest Common Divisor):
\[ \text{LCM}(X, Y) = \frac{X \cdot Y}{\text{GCD}(X, Y)} \]

Substituting \( Y \) from the first equation, we get:
\[ \text{LCM}(X, 1000 - X) = \frac{X \cdot (1000 - X)}{\text{GCD}(X, 1000 - X)} \]

### Step 3: Set up the equation using LCM
Since we know that the LCM is 8919, we can write:
\[ \frac{X \cdot (1000 - X)}{\text{GCD}(X, 1000 - X)} = 8919 \]

### Step 4: Rearranging the equation
Multiply both sides by \( \text{GCD}(X, 1000 - X) \):
\[ X \cdot (1000 - X) = 8919 \cdot \text{GCD}(X, 1000 - X) \]

### Step 5: Solve for \( X \)
Assuming \( \text{GCD}(X, 1000 - X) = 1 \) (for simplicity, as we are looking for coprime numbers), we can simplify:
\[ X \cdot (1000 - X) = 8919 \]

Expanding this gives:
\[ 1000X - X^2 = 8919 \]

Rearranging leads to:
\[ X^2 - 1000X + 8919 = 0 \]

### Step 6: Use the quadratic formula
Now we can use the quadratic formula to solve for \( X \):
\[ X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1000, c = 8919 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-1000)^2 - 4 \cdot 1 \cdot 8919 = 1000000 - 35676 = 964324 \]

Now substituting into the quadratic formula:
\[ X = \frac{1000 \pm \sqrt{964324}}{2} \]

Calculating \( \sqrt{964324} \):
\[ \sqrt{964324} = 982 \]

Thus:
\[ X = \frac{1000 \pm 982}{2} \]

Calculating the two possible values for \( X \):
1. \( X = \frac{1982}{2} = 991 \)
2. \( X = \frac{18}{2} = 9 \)

### Step 7: Find the corresponding \( Y \)
Using \( Y = 1000 - X \):
1. If \( X = 991 \), then \( Y = 1000 - 991 = 9 \)
2. If \( X = 9 \), then \( Y = 1000 - 9 = 991 \)

### Conclusion
The two numbers are \( 991 \) and \( 9 \).