Prove that if a and b are integers and a² +6² is even, then a+b is even. [Big Hint: try a contrapositive proof. Square it. See what happens.] Prove that if 5 | n, then 3n - 45 is divisible by 15.
Prove that if a and b are integers and a² +6² is even, then a+b is even. [Big Hint: try a contrapositive proof. Square it. See what happens.] Prove that if 5 | n, then 3n - 45 is divisible by 15.
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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Please follow exactly the solution steps as the answers on the 2nd and 3rd image for different questions. The way you answer my question should follow exactly those steps and procedures
![**Learning Target L5**: *I can correctly structure a careful mathematical proof including all relevant English and mathematical statements.*
Please remember the academic integrity policy that you swore to uphold. Proofs can be challenging. Just try your best. If you don’t get it this week, you have many more opportunities!
**Advice**: If you still have an N in this learning target and you’re having trouble, send me an email/message with what you have, and I’ll try to help push you in the right direction (without giving it all away). I want to help you succeed. Seriously.
1. Prove that if \( a \) and \( b \) are integers and \( a^2 + b^2 \) is even, then \( a + b \) is even. [Big Hint: try a contrapositive proof. Square it. See what happens.]
2. Prove that if 5 \(|\) \( n \), then \( 3n - 45 \) is divisible by 15.
You can use our text, videos, notes etc, posted to Teams and Blackboard. No other outside resources (including humans) are allowed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd375a7e5-ad67-4b21-afad-3fb7503a0e18%2F5687c218-8682-450d-a772-1a47da237e39%2Fbqbjflg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Learning Target L5**: *I can correctly structure a careful mathematical proof including all relevant English and mathematical statements.*
Please remember the academic integrity policy that you swore to uphold. Proofs can be challenging. Just try your best. If you don’t get it this week, you have many more opportunities!
**Advice**: If you still have an N in this learning target and you’re having trouble, send me an email/message with what you have, and I’ll try to help push you in the right direction (without giving it all away). I want to help you succeed. Seriously.
1. Prove that if \( a \) and \( b \) are integers and \( a^2 + b^2 \) is even, then \( a + b \) is even. [Big Hint: try a contrapositive proof. Square it. See what happens.]
2. Prove that if 5 \(|\) \( n \), then \( 3n - 45 \) is divisible by 15.
You can use our text, videos, notes etc, posted to Teams and Blackboard. No other outside resources (including humans) are allowed.
Transcribed Image Text:### Proofs Involving Divisibility
#### 1. Proof: Assume \(2 \mid n\) and \(3 \mid n\).
- Begin with the assumption that \(2 \mid n\) and \(3 \mid n\).
- There exist integers \(k\) and \(l\) such that \(n = 2k\) and \(n = 3l\).
- Therefore, \(2k = 3l\).
Since \(3 \mid 2\):
- It follows that \(3 \mid k\), so \(k = 3m\) for some integer \(m\).
- Thus, substituting back, \(n = 2k = 2(3m) = 6m\).
Consequently, \(6 \mid n\).
#### 2. Proof by Contraposition
- Assume that \(n\) is odd.
- There exists an integer \(k\) such that \(n = 2k + 1\).
Adding 7 to both sides:
- \(n + 7 = 2k + 1 + 7\)
- \(= 2k + 8\)
- \(= 2(k + 4)\)
Thus, \(n + 7\) is even.
Therefore, if \(n + 7\) is odd, then \(n\) is even by contraposition.
Expert Solution
Step 1: Introduction
We need to prove following statement.
(1)
If
(2)
If
We will prove the part (1) by proving its contrapositive statement.
Step by step
Solved in 4 steps with 10 images
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