If `m^n - m = (m - n)!` where `m > n > 1` and `m = n^2` then the value of `m^2 + n^2` is :

Options

A. 272
B. 90
C. 20
D. none of (a), (b), (c)

Answer

To solve the equation \( m^n - m = (m - n)! \) given that \( m > n > 1 \) and \( m = n^2 \), we will follow these steps:

### Step 1: Substitute \( m \) with \( n^2 \)
Since we know that \( m = n^2 \), we can substitute \( m \) in the equation:
\[ (n^2)^n - n^2 = (n^2 - n)! \]

### Step 2: Simplify the left side of the equation
The left side simplifies to:
\[ n^{2n} - n^2 \]

So, the equation now looks like:
\[ n^{2n} - n^2 = (n^2 - n)! \]

### Step 3: Test integer values for \( n \)
Since \( n > 1 \), we can start testing integer values for \( n \).

#### Testing \( n = 2 \):
\[ 2^{2 \cdot 2} - 2^2 = 4 - 4 = 0 \]
\[ (2^2 - 2)! = (4 - 2)! = 2! = 2 \]
This does not satisfy the equation.

#### Testing \( n = 3 \):
\[ 3^{2 \cdot 3} - 3^2 = 3^6 - 9 = 729 - 9 = 720 \]
\[ (3^2 - 3)! = (9 - 3)! = 6! = 720 \]
This satisfies the equation.

### Step 4: Find \( m \)
Now that we have \( n = 3 \), we can find \( m \):
\[ m = n^2 = 3^2 = 9 \]

### Step 5: Calculate \( m^2 + n^2 \)
Now we need to calculate \( m^2 + n^2 \):
\[ m^2 + n^2 = 9^2 + 3^2 = 81 + 9 = 90 \]

### Final Answer
Thus, the value of \( m^2 + n^2 \) is \( \boxed{90} \).

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