If `(25)_(n) xx (31)_(n) = (1015)_(n)` then the value of `(13)_(n) xx (52)_(n)` is when `n > 0` :

Options

A. `a. (626)_(4)`
B. b.`(462)_(n)`
C. c. `(716)_(n) `
D. d. `(676)_(n)`

Answer

To solve the problem step by step, we need to find the value of \( n \) such that \( (25)_n \times (31)_n = (1015)_n \), and then use that \( n \) to calculate \( (13)_n \times (52)_n \).

### Step 1: Convert the numbers from base \( n \) to decimal
1. Convert \( (25)_n \):
\[ (25)_n = 2n + 5 \]

2. Convert \( (31)_n \):
\[ (31)_n = 3n + 1 \]

3. Convert \( (1015)_n \):
\[ (1015)_n = 1n^3 + 0n^2 + 1n + 5 = n^3 + n + 5 \]

### Step 2: Set up the equation
We have:
\[ (2n + 5)(3n + 1) = n^3 + n + 5 \]

### Step 3: Expand the left-hand side
Expanding \( (2n + 5)(3n + 1) \):
\[ = 6n^2 + 2n + 15n + 5 = 6n^2 + 17n + 5 \]

### Step 4: Set the equation
Now we set the expanded left-hand side equal to the right-hand side:
\[ 6n^2 + 17n + 5 = n^3 + n + 5 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ n^3 - 6n^2 - 16n = 0 \]

### Step 6: Factor out \( n \)
Factoring out \( n \):
\[ n(n^2 - 6n - 16) = 0 \]
This gives us one solution \( n = 0 \) (which we discard since \( n > 0 \)) and a quadratic equation:
\[ n^2 - 6n - 16 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ n = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1} \]
\[ = \frac{6 \pm \sqrt{36 + 64}}{2} \]
\[ = \frac{6 \pm \sqrt{100}}{2} \]
\[ = \frac{6 \pm 10}{2} \]
This gives us two potential solutions:
\[ n = \frac{16}{2} = 8 \quad \text{and} \quad n = \frac{-4}{2} = -2 \quad (\text{discarded}) \]

### Step 8: Calculate \( (13)_n \times (52)_n \)
Now we use \( n = 8 \):
1. Convert \( (13)_n \):
\[ (13)_8 = 1 \cdot 8 + 3 = 8 + 3 = 11 \]

2. Convert \( (52)_n \):
\[ (52)_8 = 5 \cdot 8 + 2 = 40 + 2 = 42 \]

### Step 9: Multiply the two results
Now we multiply:
\[ (13)_8 \times (52)_8 = 11 \times 42 = 462 \]

### Final Result
Thus, the value of \( (13)_n \times (52)_n \) when \( n = 8 \) is \( 462 \).

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