Prove that for any integers a and b, if a b and alc, then a|(3b-5c).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question

Transcribed Image Text:**Mathematical Problem Statement:**
"Prove that for any integers \(a\) and \(b\), if \(a \mid b\) and \(a \mid c\), then \(a \mid (3b - 5c)\)."
**Explanation:**
This statement is a typical question in number theory related to divisibility. The notation \(a \mid b\) means that \(a\) divides \(b\), implying there exists an integer \(k\) such that \(b = ak\). The problem requires you to prove that under these conditions, \(a\) also divides the linear combination \(3b - 5c\).
**No graphs or diagrams are provided.**
The approach to solve this involves using the properties of divisibility and linear combinations. Here is a brief outline of how one might approach the proof:
1. Given \(a \mid b\), express \(b\) in terms of \(a\): \(b = am\) for some integer \(m\).
2. Given \(a \mid c\), express \(c\) in terms of \(a\): \(c = an\) for some integer \(n\).
3. Substitute these expressions into \(3b - 5c\):
\(3b - 5c = 3(am) - 5(an) = a(3m - 5n)\).
4. Since \(3m - 5n\) is an integer (as integers are closed under addition and multiplication), it implies that \(a \mid (3b - 5c)\).
This demonstrates that \(a\) divides \(3b - 5c\), thus completing the proof.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images

Recommended textbooks for you

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON

The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman

Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman