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## Homework Statement

For any integer n, let A(n) be the statement:

“If n 2 = 4k + 1 for some k ∈ Z, then n = 4q + 1 or 4q + 3 for some q ∈ Z.”

Use proof by contradiction to show that A(n) is true for all n ∈ Z.

## The Attempt at a Solution

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the answer sheet says that since n !=4q+1 and n != 4q+3 implies that n = 4q or n = 4q+2. I am confused why n!=4q+1 and n != 4q+3 implies that n = 4q or n = 4q+2. Why only those two examples? couldn't n = 4q+x where x = 0,2,4,5,6,7,... basically any integer that is not 1 or 3, since 'x' is defined to NOT be those with the proof via contradiction.