Prove that “If the product of two integers a and b are even then either a is even or b is even”, using the contradiction method.

Question

Prove that &#8220;If the product of two integers a and b are even then either a is even or b is even&#8221;, using the contradiction method.

Suresh Chand · Accepted Answer

We will prove the statement by contradiction. Suppose that the product of two integers a and b is even, but neither a nor b is even. This means that both a and b are odd. Let a = 2k + 1 and b = 2m + 1, where k and m are integers. Then, their product is: ab = (2k + 1)(2m + 1) = 4km + 2k + 2m + 1 = 2(2km + k + m) + 1 Since 2km + k + m is an integer, ab is odd, which contradicts the assumption that ab is even. Therefore, we can conclude that if the product of two integers a and b is even, then either a is even or b is even.

Prove that “If the product of two integers a and b are even then either a is even or b is even”, using the contradiction method.

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Prove that “If the product of two integers a and b are even then either a is even or b is even”, using the contradiction method.

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