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For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n + 1, and a = 3n + 2 for some n (and explain why these cover all possibilities).] [Another hint: (x + y)3 = x3 + 3x2y + 3xy2 + y3 for any x and y.]

For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n + 1, and a = 3n + 2 for some n (and explain why these cover all possibilities).] [Another hint: (x + y)3 = x3 + 3x2y + 3xy2 + y3 for any x and y.]

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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For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n + 1, and a = 3n + 2 for some n (and explain why these cover all possibilities).] [Another hint: (x + y)3 = x3 + 3x2y + 3xy2 + y3 for any x and y.]

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