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1. Prove that if n is an integer and n is a multiple of 2, then 4n+ 24 is a multiple of 8. 2. Show that if a = b (mod m) and b = c (mod m) where a and b are non-zero integers and m is a positive integer, then a = c (mod m). (Recommended start: write the congruences as statements of divisibility, and divisility as equalities, then go from there).

1. Prove that if n is an integer and n is a multiple of 2, then 4n+ 24 is a multiple of 8. 2. Show that if a = b (mod m) and b = c (mod m) where a and b are non-zero integers and m is a positive integer, then a = c (mod m). (Recommended start: write the congruences as statements of divisibility, and divisility as equalities, then go from there).

Advanced Engineering Mathematics
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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**Learning Target L5 (Core):** *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! 1. Prove that if \( n \) is an integer and \( n \) is a multiple of 2, then \( 4n + 24 \) is a multiple of 8. 2. Show that if \( a \equiv b \ (\text{mod} \ m) \) and \( b \equiv c \ (\text{mod} \ m) \) where \( a \) and \( b \) are non-zero integers and \( m \) is a positive integer, then \( a \equiv c \ (\text{mod} \ m) \). (Recommended start: write the congruences as statements of divisibility, and divisibility as equalities, then go from there). You can use our text, videos, notes etc, posted to Discourse. No other outside resources (including humans) are allowed.
Transcribed Image Text:**Learning Target L5 (Core):** *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! 1. Prove that if \( n \) is an integer and \( n \) is a multiple of 2, then \( 4n + 24 \) is a multiple of 8. 2. Show that if \( a \equiv b \ (\text{mod} \ m) \) and \( b \equiv c \ (\text{mod} \ m) \) where \( a \) and \( b \) are non-zero integers and \( m \) is a positive integer, then \( a \equiv c \ (\text{mod} \ m) \). (Recommended start: write the congruences as statements of divisibility, and divisibility as equalities, then go from there). You can use our text, videos, notes etc, posted to Discourse. No other outside resources (including humans) are allowed.
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We will prove sub-part 1 by method of contraposition and sub-part 2 as per given recommendation. 

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