Even and Odd Integers in Arithmetic
Introduction
Let n and m be integers, where:
n = ..., -3, -2, -1, 0, 1, 2, 3, ...
and
m = ..., -3, -2, -1, 0, 1, 2, 3, ...
An even integer is any integer that can evenly divided by 2, without remainder. Hence, if p is an even integer, then p = 2 ∙ n
An odd integer is any integer that can not be evenly divided by 2 (remainder 1). In this case: p = 2 ∙ n + 1
Addition
Adding two integers will result in an integer.
even + even = even
2 ∙ n + 2 ∙ m
= 2 ∙ (n + m)
odd + odd = even
(2 ∙ n + 1) + (2 ∙ m + 1)
= 2 ∙ n + 2 ∙ m + 2
= 2 ∙ (n + m + 1)
even + odd = odd
(2 ∙ n) + (2 ∙ m + 1)
= 2 ∙ n + 2 ∙ m + 1
= 2 ∙ (n + m) + 1
Multiplication
Multiplying two integers will result in an integer.
even × even = even
(2 ∙ n) ∙ (2 ∙ m)
= 2 ∙ (n ∙ m)
odd × odd = odd
(2 ∙ n + 1) ∙ (2 ∙ m + 1)
= 4 ∙ m ∙ n + 2 ∙ n + 2 ∙ m + 1
= 2 ∙ ( 2 ∙ m ∙ n + n + m) + 1
even × odd = even
(2 ∙ n + 1) ∙ (2 ∙ m)
= 4 ∙ m ∙ n + 2 ∙ m
= 2 ∙ (2 ∙ m ∙ n + n)
even^2 = even
(2 ∙n)^2
= 4 ∙ n^2
= 2 ∙ (2 ∙ n ∙ n)
odd^2 = odd
(2 ∙ n + 1)^2
= 4 ∙ n^2 + 4 ∙ n + 1
= 2 ∙ (2 ∙ n ∙ n + 2 ∙ n) + 1
Eddie
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