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Prove the following using the Contrapositive, Proof by Cases, or Contradiction. 1. If m and n are integers of the same parity (both even or both odd), then 7m - 3n is even. 2. Let a, b Z. If ab is even, then a is even or b is even. 3. Suppose that a, b, c € Z and a divides b + c. Show that if a does not divide b, then a does not divide c.

Prove the following using the Contrapositive, Proof by Cases, or Contradiction. 1. If m and n are integers of the same parity (both even or both odd), then 7m - 3n is even. 2. Let a, b Z. If ab is even, then a is even or b is even. 3. Suppose that a, b, c € Z and a divides b + c. Show that if a does not divide b, then a does not divide c.

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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i need help with all. this is for discrete mathematics. 

Prove the following using the Contrapositive, Proof by Cases, or Contradiction. 1. If m and n are integers of the same parity (both even or both odd), then 7m - 3n is even. 2. Let a, b € Z. If ab is even, then a is even or b is even. 3. Suppose that a, b, c € Z and a divides b + c. Show that if a does not divide b, then a does not divide c. 4. Let a, b € Z, and let p be a prime. If p does not divide ab, then p does not divide a and p does not divide b. 5. There exists no integers j, k for which 18j + 6k = 1. 6. Suppose m, n E Z. Prove that if 4 divides m² + n², then m and n are both even.
Transcribed Image Text:Prove the following using the Contrapositive, Proof by Cases, or Contradiction. 1. If m and n are integers of the same parity (both even or both odd), then 7m - 3n is even. 2. Let a, b € Z. If ab is even, then a is even or b is even. 3. Suppose that a, b, c € Z and a divides b + c. Show that if a does not divide b, then a does not divide c. 4. Let a, b € Z, and let p be a prime. If p does not divide ab, then p does not divide a and p does not divide b. 5. There exists no integers j, k for which 18j + 6k = 1. 6. Suppose m, n E Z. Prove that if 4 divides m² + n², then m and n are both even.
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