Definition 1: Let 1, y E Z. Then I is divisible by y if there exists an integer k such that I = ky. Definition 2: The product of two consecutive integers is always divisible by 2. For example, k? + 3 and k² + 4 are two consecutive integers for all integers k. Hence, its product (k2 + 3)(k² + 4) is divisibe by 2. I. Prove the following statements using the indicated proof. 1.) Let I € Z. If r is odd, then (r² + 3)2021 is even. (Direct Proof)
Definition 1: Let 1, y E Z. Then I is divisible by y if there exists an integer k such that I = ky. Definition 2: The product of two consecutive integers is always divisible by 2. For example, k? + 3 and k² + 4 are two consecutive integers for all integers k. Hence, its product (k2 + 3)(k² + 4) is divisibe by 2. I. Prove the following statements using the indicated proof. 1.) Let I € Z. If r is odd, then (r² + 3)2021 is even. (Direct Proof)
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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If this will be factored by 2.
Transcribed Image Text:Definition 1: Let 1, y E Z. Then I is divisible by y if there exists an integer k such that I = ky.
Definition 2: The product of two consecutive integers is always divisible by 2. For example, k? + 3 and
k² + 4 are two consecutive integers for all integers k. Hence, its product (k2 + 3)(k² + 4) is divisibe by 2.
I. Prove the following statements using the indicated proof.
1.) Let I € Z. If r is odd, then (r² + 3)2021 is even. (Direct Proof)
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