KatKingston9031 KatKingston9031
  • 02-10-2019
  • Mathematics
contestada

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

Respuesta :

lyndalau86
lyndalau86 lyndalau86
  • 04-10-2019

Answer:

See explanation below

Step-by-step 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.

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