Please follow exactly the solution steps as the answers on the 2nd image for a 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. LS:1. **Proof by Contradiction**: Assume, to the contrary, that $a$ and $b$ are integers so that $4a + 8b = 57$. Then: \[ 4(a + 2b) = 57 \] But since 57 isn't divisible by 4, this is impossible! Thus it must be that $a$ and $b$ are not integers. 2. **Proof by Contrapositive**: Assume that $n$ is odd. Then there exists an integer $k$ so that $n = 2k + 1$. Consider: \[ n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 \] and so $2 \nmid n^2$. Thus, if $2 \mid n^2$, we have $n$ is even, by contraposition.

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). want to help you succeed. Seriously. 1. 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.] 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.

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). want to help you succeed. Seriously. 1. 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.] 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.

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

Please follow exactly the solution steps as the answers on the 2nd image for a 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.

LS:

1. **Proof by Contradiction**:

Assume, to the contrary, that $a$ and $b$ are integers so that $4a + 8b = 57$.

Then:
\[
4(a + 2b) = 57
\]

But since 57 isn't divisible by 4, this is impossible!

Thus it must be that $a$ and $b$ are not integers.

2. **Proof by Contrapositive**:

Assume that $n$ is odd. Then there exists an integer $k$ so that $n = 2k + 1$.

Consider:
\[
n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1
\]

and so $2 \nmid n^2$.

Thus, if $2 \mid n^2$, we have $n$ is even, by contraposition.

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