If p be any odd natural number, greater than 3 then which digit will never appear as the last digit in the product of `(p^2 - 1)(p^2 + 1)` is :

Options

A. 9 and 7
B. 5 and 3
C. 1 and 5
D. all of these

Answer

To solve the problem, we need to analyze the expression \((p^2 - 1)(p^2 + 1)\) where \(p\) is any odd natural number greater than 3. We will determine the last digit of this product and identify which digit will never appear as the last digit.

### Step 1: Understanding \(p^2\) for odd \(p\)
Since \(p\) is an odd natural number, we can express \(p\) as \(p = 2k + 1\) for some integer \(k\). Squaring \(p\), we have:
\[ p^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1 \]
This means \(p^2\) is also odd, and specifically, the last digit of \(p^2\) can be determined by squaring the last digit of \(p\).

### Step 2: Finding the last digit of \(p^2\)
The last digits of odd numbers (1, 3, 5, 7, 9) and their squares are:
- Last digit of \(1^2\) is \(1\)
- Last digit of \(3^2\) is \(9\)
- Last digit of \(5^2\) is \(5\)
- Last digit of \(7^2\) is \(9\)
- Last digit of \(9^2\) is \(1\)

Thus, the possible last digits of \(p^2\) for odd \(p\) are \(1, 5, 9\).

### Step 3: Evaluating \(p^2 - 1\) and \(p^2 + 1\)
Now, we evaluate the last digits of \(p^2 - 1\) and \(p^2 + 1\):
- If the last digit of \(p^2\) is \(1\):
- Last digit of \(p^2 - 1\) is \(0\)
- Last digit of \(p^2 + 1\) is \(2\)

- If the last digit of \(p^2\) is \(5\):
- Last digit of \(p^2 - 1\) is \(4\)
- Last digit of \(p^2 + 1\) is \(6\)

- If the last digit of \(p^2\) is \(9\):
- Last digit of \(p^2 - 1\) is \(8\)
- Last digit of \(p^2 + 1\) is \(0\)

### Step 4: Finding the last digit of the product
Now we will calculate the last digit of the product \((p^2 - 1)(p^2 + 1)\):
1. If \(p^2\) ends with \(1\):
- Last digit of product = \(0 \times 2 = 0\)

2. If \(p^2\) ends with \(5\):
- Last digit of product = \(4 \times 6 = 24\) → last digit is \(4\)

3. If \(p^2\) ends with \(9\):
- Last digit of product = \(8 \times 0 = 0\)

### Step 5: Conclusion
The possible last digits of the product \((p^2 - 1)(p^2 + 1)\) are \(0\) and \(4\). The digits that can appear as the last digit are \(0\) and \(4\).

However, the digits that will never appear as the last digit are \(1, 3, 5, 7, 9\). Among the options provided, the digit that will never appear as the last digit in the product is:

**Answer: 1**