Stuck need help!The class I'm taking is computer science discrete structures. Problem is attached. please view attachment before answering. Really struggling with this concept. Thank you so much. ### Mathematical Proof Problem#### Objective:Prove each of the given statements, assuming that \( a, b, c, d, \) and \( n \) are integers with \( n > 1 \) and that - \( a \equiv c \pmod{n} \)- \( b \equiv d \pmod{n} \)#### Statement to Prove:\[ a^2 \equiv c^2 \pmod{n} \]### Explanation:Given:- Two congruences: \( a \equiv c \pmod{n} \) and \( b \equiv d \pmod{n} \).To prove: - \( a^2 \equiv c^2 \pmod{n} \)The problem requires using modular arithmetic, specifically working with congruences, to show that the square of equivalent numbers modulo \( n \) are themselves equivalent modulo \( n \).### Steps for Proof:1. From the given, \( a \equiv c \pmod{n} \) implies that \( n \) divides \( (a - c) \). That is, there exists an integer \( k \) such that: \[ a = c + kn \]2. Squaring both sides: \[ a^2 = (c + kn)^2 \]3. Expanding the squared term: \[ a^2 = c^2 + 2ckn + k^2n^2 \]4. Notice that both \( 2ckn \) and \( k^2n^2 \) are clearly divisible by \( n \).5. Thus, \( a^2 \equiv c^2 \pmod{n} \).This congruence shows that if two numbers are congruent modulo \( n \), their squares will also be congruent modulo \( n \).The approach is a simple application of algebra and understanding of modular arithmetic, illustrating basic properties of congruences.

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Answered: prove each of the given statements,…

prove each of the given statements, assuming hat a, b, c, d, and n are integers with n>1 and that = c (mod n) and b = d (mod n). a = c² (mod n)

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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