`p^(n)=(axx5)^(n)` For `p^(n)` to end with the digit zero `a-` ________________ for natural number of n.

Options

A. any natural number
B. even number
C. odd number
D. none of these

Answer

To solve the problem \( p^n = (a \times 5)^n \) for \( p^n \) to end with the digit zero, we need to analyze the expression and determine the value of \( a \).

### Step-by-Step Solution:

1. **Understanding the Problem**: We need \( p^n \) to end with the digit zero. A number ends with the digit zero if it is divisible by 10. Since \( 10 = 2 \times 5 \), we need to ensure that \( p^n \) has at least one factor of 2 and one factor of 5.

2. **Rewriting the Equation**: We start with the equation:
\[ p^n = (a \times 5)^n \]
This can be rewritten as:
\[ p^n = a^n \times 5^n \]

3. **Identifying Factors**: For \( p^n \) to end with the digit zero, it must include both factors of 2 and 5. The term \( 5^n \) provides the necessary factor of 5.

4. **Finding the Factor of 2**: To ensure that \( p^n \) has a factor of 2, \( a^n \) must provide at least one factor of 2. This means \( a \) must be an even number.

5. **Conclusion**: Therefore, for \( p^n \) to end with the digit zero, \( a \) must be an even number. The simplest even number is 2, but \( a \) can also be any other even number (like 4, 6, etc.).

### Final Answer:
Thus, \( a \) must be an even number.

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