The sum of a natural number and its reciprocal is `(65)/(8)`. Find the number.

Answer

To solve the problem, we need to find a natural number \( x \) such that the sum of the number and its reciprocal equals \( \frac{65}{8} \).

### Step 1: Set up the equation
According to the problem, we can express the relationship as:
\[ x + \frac{1}{x} = \frac{65}{8} \]

### Step 2: Eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by \( 8x \) (where \( x \) is not zero):
\[ 8x^2 + 8 = 65x \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ 8x^2 - 65x + 8 = 0 \]

### Step 4: Use the quadratic formula
Now, we will use the quadratic formula to solve for \( x \). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation \( 8x^2 - 65x + 8 = 0 \):
- \( a = 8 \)
- \( b = -65 \)
- \( c = 8 \)

### Step 5: Calculate the discriminant
First, we need to calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-65)^2 - 4 \cdot 8 \cdot 8 \]
\[ D = 4225 - 256 = 3969 \]

### Step 6: Find the square root of the discriminant
Now, we find the square root of the discriminant:
\[ \sqrt{3969} = 63 \]

### Step 7: Substitute back into the quadratic formula
Now we substitute back into the quadratic formula:
\[ x = \frac{65 \pm 63}{2 \cdot 8} \]

### Step 8: Calculate the two possible values for \( x \)
Calculating the two values:
1. \( x = \frac{65 + 63}{16} = \frac{128}{16} = 8 \)
2. \( x = \frac{65 - 63}{16} = \frac{2}{16} = \frac{1}{8} \)

### Step 9: Identify the natural number
Since we are looking for a natural number, we take:
\[ x = 8 \]

### Final Answer
The natural number is \( 8 \).

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