Question

Prove that for all positive integers x and x3 have the same odd/even status. (Hint: proof by cases)

Answer 1

See explanation below

Explanation:

Case 1) x is even ⇒ x³ is even

x = 2k for some k

⇒x³ = (2k)³= 8k³ = 2(4k³) which is even.

Case 2) x is odd⇒x³ is odd

x = 2k - 1 for some k

⇒x³ = (2k - 1)³ = (2k)³- 3(2k)²(1) + 3(2k)(1)² -(1)³ = 8k³- 12k² + 6 - 1 = 2 (4k³ - 6k² + 3) - 1 which is odd.