If a number 'n' can exactly, divide `(5^14-1)` then 'n' can necessarily divide:

Options

A. (5^28-1)`
B. `(5^42-1)`
C. `(5^21-1)`
D. both (a) and(b)

Answer

To solve the problem, we need to determine what other numbers can be divided by \( n \) if \( n \) can exactly divide \( 5^{14} - 1 \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \( 5^{14} - 1 \). We know that if \( n \) divides \( 5^{14} - 1 \), then \( n \) must also divide any expression that can be derived from \( 5^{14} - 1 \).

2. **Using the Difference of Squares**:
We can factor \( 5^{14} - 1 \) using the difference of squares:
\[ 5^{14} - 1 = (5^7 - 1)(5^7 + 1) \]

3. **Further Factoring**:
We can further factor \( 5^7 - 1 \):
\[ 5^7 - 1 = (5^{3.5} - 1)(5^{3.5} + 1) \]
However, since \( 3.5 \) is not an integer, we will not factor it this way. Instead, we can factor \( 5^7 - 1 \) as:
\[ 5^7 - 1 = (5 - 1)(5^6 + 5^5 + 5^4 + 5^3 + 5^2 + 5 + 1) \]

4. **Finding the Divisibility**:
Since \( n \) divides \( 5^{14} - 1 \), we can use the property of divisibility in powers. If \( n \) divides \( 5^{14} - 1 \), then \( n \) also divides \( 5^{28} - 1 \) because:
\[ 5^{28} - 1 = (5^{14} - 1)(5^{14} + 1) \]
Therefore, \( n \) must also divide \( 5^{28} - 1 \).

5. **Conclusion**:
Thus, if \( n \) divides \( 5^{14} - 1 \), it necessarily divides \( 5^{28} - 1 \).

### Final Answer:
If a number \( n \) can exactly divide \( 5^{14} - 1 \), then \( n \) can necessarily divide \( 5^{28} - 1 \).