A. prove: let a, b, and c be integers. if (a. b) c, then a c. suppose a, b, c are integers with (a. b) (2.b) ] c. then c=( c. then c = ).2.b) for some integer k, so c=( . (a. b) for some integer k, so c= a, so a c. b. prove the following statement by contraposition. let x be an integer. if x2 + x + 1 is even, then x is odd. make sure that your proof makes appropriate use of definitions 1.5 and 1.6. let x be an integer. suppose that is even. by definition 1.5, x = 2k for some integer k. then x2 + x + 1 = )k + 1 =( enter an exact number. (2k2 + k) + 1 is , by definition 1.6. enter an exact number. odd even c. write a proof by contradiction of the following. let x and y be integers. if x and y satisfy the equation 3x + 5y = 145 then at least one of x and y is odd. let x and y be integers that satisfy the equation 3x + 5y = 145. suppose, to the contrary, that both x and y are even. then x = 2k' for some integer kį and y = 262 for some integer k2. then 145 = 3x + 5y = ( 6 )k+ ( 10 )k2 = 2 (3k\ + 5k2) is odd . but 145 = 2.0 +1 is also odd enter an exact number. this contradicts axiom 1.2.