Assertion (A): `6^(n)` ends with the digit zero where n is a natural number.
Reason (R):Any number ends with digit zero, if its prime factor is of the form `2^(m)xx5^(n)`, where m,n are natural numbers.

Options

A. Both A and R are true and R is the correct explanation for A .
B. Both A and Rare true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the given assertion and reason, we will analyze each statement step by step.

### Step 1: Analyze the Assertion (A)
The assertion states that \(6^n\) ends with the digit 0, where \(n\) is a natural number.

- **Prime Factorization of 6**:
\[ 6 = 2 \times 3 \]
- **Raising to the power of n**:
\[ 6^n = (2 \times 3)^n = 2^n \times 3^n \]

### Step 2: Determine the Conditions for Ending with 0
A number ends with the digit 0 if its prime factorization includes at least one factor of 2 and one factor of 5.

- **Check for the presence of 5**:
In the prime factorization \(2^n \times 3^n\), we see that there is no factor of 5 present.

### Conclusion for Assertion (A)
Since \(6^n\) does not include the factor 5, it cannot end with the digit 0. Therefore, the assertion (A) is **false**.

### Step 3: Analyze the Reason (R)
The reason states that any number ends with the digit 0 if its prime factorization is of the form \(2^m \times 5^n\), where \(m\) and \(n\) are natural numbers.

- This statement is true because for a number to end with 0, it must have at least one factor of 2 and one factor of 5.

### Conclusion for Reason (R)
The reason (R) is **true**.

### Final Conclusion
- Assertion (A) is **false**.
- Reason (R) is **true**.
- Therefore, the correct answer is that A is false, but R is true.

---