Prove the following for Integers x and y, (17 | 10x + 8y) = (17 | 3x + 16y)
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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Question
![**Proof for Integer Expression Equivalence**
**Objective**: Prove the following for integers \( x \) and \( y \):
\[
(17 \mid 10x + 8y) \equiv (17 \mid 3x + 16y)
\]
**Explanation**:
This expression involves proving the equivalence of two modular conditions. It uses modular arithmetic notation, where \( a \mid b \) indicates that \( a \) divides \( b \), which in modular arithmetic is denoted as \( b \equiv 0 \pmod{a} \).
**Steps**:
1. Analyze both sides of the equivalence:
- Left side: \( 17 \mid (10x + 8y) \)
- Right side: \( 17 \mid (3x + 16y) \)
2. Transforming the expressions to show that they produce equivalent congruence relations when divided by 17.
3. Expand and simplify both sides to determine if a transformation or substitution exists to prove their equality under modulo 17.
**Solution**: A detailed proof approach would involve algebraic manipulation and verification that the expressions on both sides remain congruent for all integer values of \( x \) and \( y \).
For educational purposes, this example highlights key concepts in congruences, linear combinations, and logical equivalences in modular arithmetic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F233b4b28-2634-4ce7-bb56-644fea0dac02%2Fed83b527-8994-45b2-8ec1-5519795d113c%2Fjtttdl7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Proof for Integer Expression Equivalence**
**Objective**: Prove the following for integers \( x \) and \( y \):
\[
(17 \mid 10x + 8y) \equiv (17 \mid 3x + 16y)
\]
**Explanation**:
This expression involves proving the equivalence of two modular conditions. It uses modular arithmetic notation, where \( a \mid b \) indicates that \( a \) divides \( b \), which in modular arithmetic is denoted as \( b \equiv 0 \pmod{a} \).
**Steps**:
1. Analyze both sides of the equivalence:
- Left side: \( 17 \mid (10x + 8y) \)
- Right side: \( 17 \mid (3x + 16y) \)
2. Transforming the expressions to show that they produce equivalent congruence relations when divided by 17.
3. Expand and simplify both sides to determine if a transformation or substitution exists to prove their equality under modulo 17.
**Solution**: A detailed proof approach would involve algebraic manipulation and verification that the expressions on both sides remain congruent for all integer values of \( x \) and \( y \).
For educational purposes, this example highlights key concepts in congruences, linear combinations, and logical equivalences in modular arithmetic.
Expert Solution
Step 1
17 | 10x + 8y where x and y are integers
=> 10x + 8y = 17m for some integer m
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