lementary number theory and methods of proof

1. disprove: For all real numbers a and b, if a <b then a^2 <b^2

2. prove: the product of any even integer and any integer is even.

3. prove: for all intehers a,b and c, if a|b and a|c then a|(b-c)

4-prove that the square of any integer has the form 3k or 3k+1 for some integer k.

hint: use the quotient-remainder theorem to show that the integer can be written in one of three forms.

5. prove by contradiction: there is no greatest negative real number.

 

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