Prove the following biconditional statements Prove that 3 di

Prove the following biconditional statements. Prove that 3 divides 2n^2 + 1 if and only if 3 does not divide n.

Given:

3 does not divide n

Case 1: When n is divided by 3, Remainder = 1

Then,

n = 3x + 1, where x is an integer.

So,

the other condition:

2n² + 1 becomes

2(3x+1)² + 1

= 2(9x² + 6x + 1) + 1

= 18 x² + 12 x + 3.

It is seen that the coefficient of each of the 3 terms is divisible by 3.

So, 2n²+1 is divisible by 3.

Case 2: When n is divided by 3, remainder = 2.

Then

n = 3x + 2, where x is an integer.

So, the other condition:

2n² + 1 becomes:

2(3x+2)² + 1

= 2(9x² + 12x + 4) + 1

= 18 x² + 24x + 2 + 1

It is seen the coefficient of each of the 3 terms is divisible by 3.

So, 2n² + 1 is divisible by 3.