Question

question in photo i want solution

Answer 1

Let n be an even positive integer.
When n = 2q 
In this case , we have 
n2 - n = (2q)2 - 2q = 4q2 - 2q = 2q (2q - 1 )
n2 - n = 2r , where r = q (2q - 1)
n2 - n is divisible by 2 .

Case ii: Let n be an odd positive integer.
When n = 2q + 1
In this case 
n2 -n = (2q + 1)2 - (2q + 1)= (2q +1) ( 2q+1 -1)= 2q (2q + 1)
n2 - n = 2r , where  r = q (2q + 1)
n2 - n is divisible by 2.

∴  n 2 - n is divisible by 2 for every integer n

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