if n is not divisible by 3, then n 2 + 2 is divisible by 3

Question

Prove or disprove the statement 

if n is not divisible by 3, then n 2 + 2 is divisible by 3

Answer 1

By Euclidean division, the remainder of n divided by 3 is 1 or 2, i.e. n can be written as

 

n = 3k+r,

 

where r is 1 or 2 and k is an integer.

 

(3k+1)²+2 = 9k²+6k+1+2

= 3(3k²+2k+1)

 

A similar argument shows that (3k+2)²+2 is divisible by 3.