Question #337131

Show that if an integer n is not divisible by 3, then n2 – 1 must be a multiple of 3


1
Expert's answer
2022-05-06T05:53:23-0400

If nn is not divisible by 3, then either n=3k+1,kZn=3k+1, k\in \Z or n=3m+2,n=3m+2, mZ.m\in \Z.

Then


n21=(3k+1)21=9k2+6kn^2-1=(3k+1)^2-1=9k^2+6k

=3(3k2+2k),kZ=3(3k^2+2k), k\in \Z

Let p=3k2+2k.p=3k^2+2k. We know that pp is an integer because kk is an integer. Therefore n21n^2-1 equals 3 times some integer. So n21n^2-1 is a multiple of 3.

Or


n21=(3m+2)21=9m2+12m+3n^2-1=(3m+2)^2-1=9m^2+12m+3

=3(3m2+4m+1),mZ=3(3m^2+4m+1), m\in \Z

Let q=3m2+4m+1.q=3m^2+4m+1. We know that qq is an integer because mm is an integer. Therefore n21n^2-1 equals 3 times some integer. So n21n^2-1 is a multiple of 3.

Therefore if an integer nn is not divisible by 3, then n21n^2 – 1 must be a multiple of 3.


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