Prove that the product of an odd integer and an even integer is an even integer.

Answer 1

Definition 1: An integer n is even if there is an integer m such that n = 2m.

Definition 2: An integer n is odd if there is an integer m such that n = 2m + 1.

Claim: The product of an odd integer and an even integer is an even integer.

Let k be an odd integer and let x be an even integer.

Since k is an odd integer, there exists an integer j such that k = 2j + 1.

Since x is an even integer, there exists an integer y such that x=2y.

Thus, k*x = (2j+1)(2y) = 4jy + 2y = 2(2jy + y).

Since j and y are integers, 2jy+y is an integer, so 2(2jy+y) is an even integer by Definition 2.

Thus, k*x is an even integer and the product of an odd integer and an even integer is an even integer.