The HCF of `(a -1) (a^(3) + m) and (a +1) (a^(3) -n) and (a +1) (a^(2) -n) " is " a^(2) -1`, then the value of m and n are ______

Answer

To find the values of \( m \) and \( n \) given that the HCF of the expressions \( (a - 1)(a^3 + m) \), \( (a + 1)(a^3 - n) \), and \( (a + 1)(a^2 - n) \) is \( a^2 - 1 \), we can follow these steps:

### Step 1: Understand the HCF
The HCF of the three expressions is given as \( a^2 - 1 \). We know that \( a^2 - 1 \) can be factored as \( (a - 1)(a + 1) \).

### Step 2: Analyze the first expression
The first expression is \( (a - 1)(a^3 + m) \). For \( a^2 - 1 \) to be a factor of this expression, \( a^3 + m \) must also be divisible by \( a + 1 \) (since \( a^2 - 1 = (a - 1)(a + 1) \)).

### Step 3: Set up the condition for divisibility
For \( a^3 + m \) to be divisible by \( a + 1 \), we can substitute \( a = -1 \):
\[ (-1)^3 + m = -1 + m = 0 \implies m = 1 \]

### Step 4: Analyze the second expression
The second expression is \( (a + 1)(a^3 - n) \). For \( a^2 - 1 \) to be a factor, \( a^3 - n \) must also be divisible by \( a - 1 \).

### Step 5: Set up the condition for divisibility
For \( a^3 - n \) to be divisible by \( a - 1 \), we substitute \( a = 1 \):
\[ (1)^3 - n = 1 - n = 0 \implies n = 1 \]

### Step 6: Conclusion
Thus, we have found that:
\[ m = 1 \quad \text{and} \quad n = 1 \]

### Final Answer
The values of \( m \) and \( n \) are:
\[ m = 1, \quad n = 1 \]