Given
If n is not divisible by 3 then n2-1 is divisible by 3 ∀ n ∈ Z
Proof
For the value of n, we have 3 choices
Case 1 : n % 3 = 1
Let,
P = (n2-1) % 3
P = ( ( n2 ) % 3 – 1 % 3 ) % 3
P = ( ( n % 3 ) ( n % 3 ) – 1 ) % 3
P = ( 1 * 1 – 1 ) % 3
P = 0 % 3
P = 0
Therefore, n2-1 is divisible by 3 if n % 3 = 1
Case 2 : n % 3 = 2
Let,
P = (n2-1) % 3
P = ( ( n2 ) % 3 – 1 % 3 ) % 3
P = ( ( n % 3 ) ( n % 3 ) – 1 ) % 3
P = ( 2 * 2 – 1 ) % 3
P = 3 % 3
P = 0
Therefore, n2-1 is divisible by 3 if n % 3 = 2
Case 2 : n % 3 = 0
P = ( n2 – 1 ) % 3
P = ( ( n2 ) % 3 – 1 % 3 ) % 3
P = ( ( n % 3 ) ( n % 3 ) – 1 ) % 3
P = ( 0 * 0 – 1 ) % 3
P = -1 % 3
P = 2
Therefore, n2+2 is not divisible by 3 if n % 3 = 0
From the above cases, we can conclude that If n is not divisible by 3 then n2-1 is divisible by 3 ∀ n ∈ Z.