- #1
nicnicman
- 136
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Hello all I've been practicing proofs and would like to know if I'm on the right track. Here it is:
If the sum of 3x + 3y is an odd number then x and y are different parities.
Proof: Let x and y be two integers with opposite parity. Without loss of generality, suppose x is even and y is odd:
x = 2m
y = 2n + 1
Then:
3x + 3y = 3(2m) + 3(2n + 1) = 6m + 6n + 1 = 2(3m + 3n) + 1
Since 2(3m + 3n) has a factor of 2 it is even. When 1 is added, 2(3m + 3n) + 1 is odd. Therefore, 3x + 3y is odd and x and y have opposite parities.
Is this enough or do I need more?
If the sum of 3x + 3y is an odd number then x and y are different parities.
Proof: Let x and y be two integers with opposite parity. Without loss of generality, suppose x is even and y is odd:
x = 2m
y = 2n + 1
Then:
3x + 3y = 3(2m) + 3(2n + 1) = 6m + 6n + 1 = 2(3m + 3n) + 1
Since 2(3m + 3n) has a factor of 2 it is even. When 1 is added, 2(3m + 3n) + 1 is odd. Therefore, 3x + 3y is odd and x and y have opposite parities.
Is this enough or do I need more?