If two positive integers differ by 4 and sum of their reciprocals is `(10)/(21)`, then one of the number is

Options

A. 1
B. 5
C. 3
D. 21

Answer

To solve the problem, we need to find two positive integers that differ by 4 and whose reciprocals sum to \( \frac{10}{21} \).

### Step-by-Step Solution:

1. **Define the Variables**:
Let the two positive integers be \( x \) and \( y \). According to the problem, we know:
\[ x - y = 4 \quad \text{(1)} \]

2. **Sum of Reciprocals**:
The sum of their reciprocals is given as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{10}{21} \quad \text{(2)} \]

3. **Rewrite Equation (1)**:
From equation (1), we can express \( x \) in terms of \( y \):
\[ x = y + 4 \quad \text{(3)} \]

4. **Substitute into Equation (2)**:
Substitute equation (3) into equation (2):
\[ \frac{1}{y + 4} + \frac{1}{y} = \frac{10}{21} \]

5. **Combine the Left Side**:
The left side can be combined:
\[ \frac{y + (y + 4)}{y(y + 4)} = \frac{10}{21} \]
Simplifying the numerator:
\[ \frac{2y + 4}{y(y + 4)} = \frac{10}{21} \]

6. **Cross-Multiply**:
Cross-multiplying gives:
\[ 21(2y + 4) = 10y(y + 4) \]

7. **Expand Both Sides**:
Expanding both sides:
\[ 42y + 84 = 10y^2 + 40y \]

8. **Rearranging the Equation**:
Rearranging gives us a quadratic equation:
\[ 10y^2 - 2y - 84 = 0 \quad \text{(4)} \]

9. **Using the Quadratic Formula**:
To solve the quadratic equation (4), we use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 10 \), \( b = -2 \), and \( c = -84 \).

10. **Calculate the Discriminant**:
Calculate the discriminant:
\[ b^2 - 4ac = (-2)^2 - 4 \cdot 10 \cdot (-84) = 4 + 3360 = 3364 \]

11. **Substituting into the Formula**:
Substitute back into the formula:
\[ y = \frac{2 \pm \sqrt{3364}}{20} \]
Since \( \sqrt{3364} = 58 \):
\[ y = \frac{2 \pm 58}{20} \]

12. **Finding the Values of \( y \)**:
This gives two potential solutions:
\[ y = \frac{60}{20} = 3 \quad \text{and} \quad y = \frac{-56}{20} = -2.8 \]
Since \( y \) must be a positive integer, we take \( y = 3 \).

13. **Finding \( x \)**:
Now, substitute \( y \) back into equation (3) to find \( x \):
\[ x = y + 4 = 3 + 4 = 7 \]

### Final Answer:
One of the numbers is \( 3 \).