Prove that n2-n is divisble by 2 for every positive integers n

1.	Prove that n2-n is divisble by 2 for every positive integers n
Answer» Any positive integer is of the form 2q or 2q + 1, for some integer q.{tex}\\therefore{/tex} When n = 2q{tex}\\style{font-family:Arial}{n^2\\;-\\;n\\;=\\;n(n\\;-\\;1)\\;=\\;2q(2q\\;-\\;1)=\\;2m,}{/tex}where m = q(2q - 1) ( m is any integer)This is divisible by 2When n = 2q + 1{tex}\\style{font-family:Arial}{\\begin{array}{l}n^2\\;-\\;n\\;=\\;n(n\\;-\\;1)\\;=\\;(2q\\;+\\;1)(2q+1-1)\\\\=2q(2q+1)\\end{array}}{/tex}= 2m, when m = q(2q + 1) ( m is any integer)which is divisible by 2.Hence, n2 - n is divisible by 2 for every positive integer n.

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