MathCS.org - Real Analysis: Examples 1.1.5(a):

Examples 1.1.5(a):

Prove that when two even integers are multiplied, the result is an even integer, and when two odd integers are multiplied, the result is an odd integer.

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To prove this we first need to know what exactly an even and odd integer is:

an integer x is even if x = 2n for some integer n
an integer x is odd if x = 2n + 1 for some integer n

Now that we have a precise definition, the actual proof is easy: Take x and y two even numbers. Then

x = 2n for some integer n
y = 2m for some integer m

Multiplying these numbers together we get

xy = (2n)(2m) = 4 nm = 2 (2nm) = 2 k

where k = 2nm. Hence, xy is again even.

If x and y are two odd numbers, then

x = 2n + 1 for some integer n
y = 2m + 1 for some integer m

Multiplying these numbers together we get

xy = (2n+1)(2m+1) = 4nm + 2(n + m) + 1 = 2 (2nm + n + m) + 1 = 2k + 1

where k = 2nm + n + m. Hence, xy is again odd.

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Examples 1.1.5(a):