Prove that ifn is odd, then n=4k+1 for some k∈Z or n=4k+3 for some k∈Z

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Samuel Hooper · Accepted Answer

Here&#039;s a basic outline:Proposition: If n is odd, then n=4k+1 for some       k    ∈    ℤ   or n=4k+3 for some       k    ∈    ℤ  .Proof: Let       ∴    n   where n is odd. Divide n by 4.Then, by division algorithm, R=0,1,2, or 3 (remainder).Case 1: R=0. If the remainder is 0, then n=4k=2(2k).      ∴    n   is evenCase 2: R=1. If the remainder is 1, then n=4k+1.      ∴    n   is odd.Case 3: R=2. If the remainder is 2, then n=4k+2=2(2k+1).      ∴    n   is even.

Davirnoilc · Accepted Answer

Case 4: R=3. If the remainder is 3, then n=4k+3.      ∴    n   is odd.      ∴    n    =    4    k    +    1                or              n    =    4    k    +    3   if n is odd

Prove that ifn is odd, then n=4k+1 for some k in ZZ or n=4k+3 for some k in ZZ

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