Let x be an integer. If x is odd, then x^2 is odd

Proof: Let x be an odd number. This means that x = 2n + 1 where n is an integer.

x^2 = (2n + 1)^2 = (2n + 1)(2n + 1)

x^2 = 4n^2 + 4n + 1

x^2 = 2(2n^2 + 2n) + 1

2(2n^2 + 2n) is an even number since 2 multiplied by any integer is even

So adding 1 is an odd number

Proof: Let x be an odd number. This means that x = 2n + 1 where n is an integer.

__Squaring x, we get:__x^2 = (2n + 1)^2 = (2n + 1)(2n + 1)

x^2 = 4n^2 + 4n + 1

x^2 = 2(2n^2 + 2n) + 1

2(2n^2 + 2n) is an even number since 2 multiplied by any integer is even

So adding 1 is an odd number

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