Problem 1: Let n N so that n ≥ 2. Assume that for a, b, a', b' = Z, a = b (mod n) and a' = b' (mod n), Prove that (a + a') = (b + b') (mod n) and that aa' = bb' (mod n) and that for any k € N+ that ak = bk (mod n).
Problem 1: Let n N so that n ≥ 2. Assume that for a, b, a', b' = Z, a = b (mod n) and a' = b' (mod n), Prove that (a + a') = (b + b') (mod n) and that aa' = bb' (mod n) and that for any k € N+ that ak = bk (mod n).
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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![**Problem 1:**
Let \( n \in \mathbb{N} \) so that \( n \geq 2 \). Assume that for \( a, b, a', b' \in \mathbb{Z} \), \( a \equiv b \pmod{n} \) and \( a' \equiv b' \pmod{n} \), Prove that
\[ (a + a') \equiv (b + b') \pmod{n} \]
and that
\[ aa' \equiv bb' \pmod{n} \]
and that for any \( k \in \mathbb{N}^{+} \) that \( a^k \equiv b^k \pmod{n} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd84f55b1-6045-46a9-9610-9a34a878a241%2Fdc64ceb9-d63a-4deb-a925-bbf6b95cb17e%2Fpq573fl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1:**
Let \( n \in \mathbb{N} \) so that \( n \geq 2 \). Assume that for \( a, b, a', b' \in \mathbb{Z} \), \( a \equiv b \pmod{n} \) and \( a' \equiv b' \pmod{n} \), Prove that
\[ (a + a') \equiv (b + b') \pmod{n} \]
and that
\[ aa' \equiv bb' \pmod{n} \]
and that for any \( k \in \mathbb{N}^{+} \) that \( a^k \equiv b^k \pmod{n} \).
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