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Prove using the notion of without loss of generality that 5x + 5y is an odd...

Prove using the notion of without loss of generality that 5x + 5y is an odd integer when x and y are integers of opposite parity.

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Answer #1

Without loss of genrality Assume that x is odd and y is even.

Let x=2k+1 , where k is some integer

Let y=2i , where i is some integer

5x+5y = 5(2k+1)+5(2i)

            = 10k+5+10i

            = 10k+10i+4+1

            = 2(5k+5i+2)+1

we can label l = 5k+5i+2 and we know l is an integer.

Then from 5x+5y = 2l+1 we know it is an odd integer

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