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Suppose p is a prime and a and k are positive integers. Prove: If plak, then pklak.

Suppose p is a prime and a and k are positive integers. Prove: If plak, then pklak.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Mathematical Statement:** Suppose \( p \) is a prime and \( a \) and \( k \) are positive integers. Prove the following statement: If \( p \mid a^k \), then \( p^k \mid a^{k} \). **Explanation:** - **Definitions:** - \( p \mid a^k \) means that \( p \) divides \( a^k \), indicating that \( a^k \) is a multiple of \( p \). - \( p^k \mid a^{k} \) means that \( p^k \) divides \( a^k \), indicating that \( a^k \) is a multiple of \( p^k \). **Structure of the Proof:** To prove this statement, you need to show that if a prime \( p \) divides \( a^k \), it must divide \( a \) (because \( p \) is a prime factor of \( a^k \)). By repeating this logic \( k \) times, you can conclude that \( p^k \) divides \( a^k \).
Transcribed Image Text:**Mathematical Statement:** Suppose \( p \) is a prime and \( a \) and \( k \) are positive integers. Prove the following statement: If \( p \mid a^k \), then \( p^k \mid a^{k} \). **Explanation:** - **Definitions:** - \( p \mid a^k \) means that \( p \) divides \( a^k \), indicating that \( a^k \) is a multiple of \( p \). - \( p^k \mid a^{k} \) means that \( p^k \) divides \( a^k \), indicating that \( a^k \) is a multiple of \( p^k \). **Structure of the Proof:** To prove this statement, you need to show that if a prime \( p \) divides \( a^k \), it must divide \( a \) (because \( p \) is a prime factor of \( a^k \)). By repeating this logic \( k \) times, you can conclude that \( p^k \) divides \( a^k \).
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