The sum of two numbers is 56 and their LCM is 105. Find the numbers.

Options

A. 40, 16
B. 45, 11
C. 21, 35
D. 31, 25

Answer

To find the two numbers given that their sum is 56 and their LCM is 105, we can follow these steps:

### Step 1: Let the two numbers be \( x \) and \( y \).
We know that:
\[ x + y = 56 \]
\[ \text{LCM}(x, y) = 105 \]

### Step 2: Express one variable in terms of the other.
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 56 - x \]

### Step 3: Use the relationship between LCM and HCF.
The relationship between LCM and HCF (Highest Common Factor) states:
\[ \text{LCM}(x, y) \times \text{HCF}(x, y) = x \times y \]
Let \( \text{HCF}(x, y) = h \). Then we can write:
\[ 105 \times h = x \times y \]

### Step 4: Substitute \( y \) from Step 2 into the equation.
Substituting \( y \) gives us:
\[ x \times (56 - x) = 105h \]
This simplifies to:
\[ 56x - x^2 = 105h \]

### Step 5: Find possible values of \( h \).
Since \( h \) is a common factor of both \( x \) and \( y \), it must also be a divisor of their LCM (105). The divisors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. We will test these values.

### Step 6: Test possible values of \( h \).
1. **If \( h = 1 \)**:
\[ 56x - x^2 = 105 \times 1 \implies x^2 - 56x + 105 = 0 \]
The discriminant \( D = 56^2 - 4 \times 1 \times 105 = 3136 - 420 = 2716 \) (not a perfect square).

2. **If \( h = 3 \)**:
\[ 56x - x^2 = 105 \times 3 \implies x^2 - 56x + 315 = 0 \]
The discriminant \( D = 56^2 - 4 \times 1 \times 315 = 3136 - 1260 = 1876 \) (not a perfect square).

3. **If \( h = 5 \)**:
\[ 56x - x^2 = 105 \times 5 \implies x^2 - 56x + 525 = 0 \]
The discriminant \( D = 56^2 - 4 \times 1 \times 525 = 3136 - 2100 = 1026 \) (not a perfect square).

4. **If \( h = 7 \)**:
\[ 56x - x^2 = 105 \times 7 \implies x^2 - 56x + 735 = 0 \]
The discriminant \( D = 56^2 - 4 \times 1 \times 735 = 3136 - 2940 = 196 \) (perfect square).

### Step 7: Solve the quadratic equation.
For \( h = 7 \):
\[ x^2 - 56x + 735 = 0 \]
Using the quadratic formula:
\[ x = \frac{56 \pm \sqrt{196}}{2} = \frac{56 \pm 14}{2} \]
Calculating the two possible values:
\[ x = \frac{70}{2} = 35 \quad \text{and} \quad x = \frac{42}{2} = 21 \]

### Step 8: Find \( y \).
Using \( y = 56 - x \):
- If \( x = 35 \), then \( y = 56 - 35 = 21 \).
- If \( x = 21 \), then \( y = 56 - 21 = 35 \).

### Conclusion:
The two numbers are **21 and 35**.