true or false? Let x be a prime power (a number of the form pe where p is a prime number and e is a positive natural). Then it is possible that x divides a product ab, where a and b are each positive naturals, but x does not divide either a or b.

**False**.

If \( x \) is a prime power, say \( x = p^e \) where \( p \) is a prime and \( e \) is a positive integer, then for \( x \) to divide the product \( ab \), it must be that at least one of \( a \) or \( b \) contains enough factors of \( p \) to contribute to the entire factor \( p^e \). 

Here’s why:

1. **Prime Factorization Requirement**: According to the properties of divisibility, if \( x \) divides \( ab \), then the exponent of \( p \) in \( ab \) must be at least \( e \). That means either \( a \) or \( b \) must contain a factor of \( p \) raised to at least the power \( e \) (or higher).

2. **Contradiction Argument**: Suppose \( x \) divides \( ab \), but \( x \) does not divide either \( a \) or \( b \). This implies that neither \( a \) nor \( b \) contain \( p \) raised to at least the power \( e \). This would contradict the requirement for \( x = p^e \) to divide \( ab \).

Therefore, it is **impossible** for \( x \) to divide \( ab \) without dividing either \( a \) or \( b \).

### Summary
If \( x \) is a prime power \( p^e \), and \( x \) divides \( ab \), then \( x \) must divide at least one of \( a \) or \( b \). So, the statement is **false**.

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Would you like a more detailed explanation or have any follow-up questions? Here are five related questions that might help:

1. What if \( x \) is a product of two distinct prime powers instead of a single prime power?
2. How does this result extend to composite numbers instead of prime powers?
3. Why is the property of prime numbers important for this conclusion?
4. How would the proof differ if \( x \) were a prime power with a different exponent?
5. Can you explain what happens when \( a \) and \( b \) are relatively prime?

**Tip:** Understanding the properties of prime factorization and divisibility is crucial in many number theory problems. Always consider the role of the prime factors in each number!

Let x be a prime power (a number of the form pe where p is a prime number and e is a positive natural). Then it is possible that x divides a product ab, where a and b are each positive naturals, but x does not divide either a or b.

Explore the divisibility properties of prime powers with this problem. Can a prime power divide a product without dividing either factor? Learn about the fundamental theorem of arithmetic and the role of prime factorization in divisibility. Suitable for undergraduate students studying number theory or discrete mathematics.

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