If `(225)_(n)= (89)_(10)` then what will be the value of `n` ?

Options

A. 8
B. 7
C. 6
D. 5

Answer

To solve the equation \( (225)_n = (89)_{10} \), we need to convert the number \( 225 \) from base \( n \) to base \( 10 \) and set it equal to \( 89 \).

### Step-by-Step Solution:

1. **Understand the representation of the number in base \( n \)**:
The number \( (225)_n \) can be expressed in decimal (base 10) as:
\[ 2n^2 + 2n + 5 \]
Here, \( 2 \) is in the hundreds place (which corresponds to \( n^2 \)), \( 2 \) is in the tens place (which corresponds to \( n^1 \)), and \( 5 \) is in the units place (which corresponds to \( n^0 \)).

2. **Set up the equation**:
We know that this must equal \( 89 \):
\[ 2n^2 + 2n + 5 = 89 \]

3. **Rearrange the equation**:
Subtract \( 89 \) from both sides to set the equation to zero:
\[ 2n^2 + 2n + 5 - 89 = 0 \]
Simplifying this gives:
\[ 2n^2 + 2n - 84 = 0 \]

4. **Divide the entire equation by 2**:
To simplify, divide every term by \( 2 \):
\[ n^2 + n - 42 = 0 \]

5. **Factor the quadratic equation**:
We need to factor \( n^2 + n - 42 \). We look for two numbers that multiply to \( -42 \) and add to \( 1 \) (the coefficient of \( n \)):
\[ n^2 + 7n - 6n - 42 = 0 \]
This can be factored as:
\[ (n + 7)(n - 6) = 0 \]

6. **Solve for \( n \)**:
Set each factor to zero:
\[ n + 7 = 0 \quad \text{or} \quad n - 6 = 0 \]
This gives:
\[ n = -7 \quad \text{or} \quad n = 6 \]

7. **Determine the valid base**:
Since a base cannot be negative, we discard \( n = -7 \). Thus, the only valid solution is:
\[ n = 6 \]

### Final Answer:
The value of \( n \) is \( 6 \).