Given
2n2 + n + 1 is not divisible by 3 ∀ n ∈ Z
Proof
We will prove the given statement by direct method.
For each value n, we have three choices
CASE 1 : n % 3 = 0
S = 2n2 + n + 1
Taking MOD 3 on both sides
S % 3 = ( 2n2 + n + 1 ) % 3
= ( 2 ( n % 3 ) ( n % 3 ) + ( n % 3 ) + ( 1 % 3 ) ) % 3
= ( 2(0)(0) + (0) + 1 ) % 3
= 1 % 3
= 1
Therefore, the given statement is not divisible by 3 if n % 3 = 0.
CASE 1 : n % 3 = 1
S = 2n2 + n + 1
Taking MOD 3 on both sides
S % 3 = ( 2n2 + n + 1 ) % 3
= ( 2 ( n % 3 ) ( n % 3 ) + ( n % 3 ) + ( 1 % 3 ) ) % 3
= ( 2(1)(1) + (1) + 1 ) % 3
= 4 % 3
= 1
Therefore, the given statement is not divisible by 3 if n % 3 = 1.
CASE 1 : n % 3 = 2
S = 2n2 + n + 1
Taking MOD 3 on both sides
S % 3 = ( 2n2 + n + 1 ) % 3
= ( 2 ( n % 3 ) ( n % 3 ) + ( n % 3 ) + ( 1 % 3 ) ) % 3
= ( 2(2)(2) + (2) + 1 ) % 3
= 11 % 3
= 2
Therefore, the given statement is not divisible by 3 if n % 3 = 2.
From the above 3 cases, we can conclude that the given statement 2n2 + n + 1 is not divisible by 3 ∀ n ∈ Z. Hence Proved.