Mathematical Proofs

Edexcel IAL WMA12/01/P2/Jan 2023/Q10 (Proof: Exhaustion)

A student was asked to prove by exhaustion that​​ 

if n is an integer then​​ 2n2 + n + 1​​ is not divisible by 3

The start of the student’s​​ proof is shown in the box below.

Complete this proof.​​ 

(4)

SOLUTION

Let us first consider all even numbers:​​ n=3k+1.

2n2 + n + 1=23k+12+3k+1+1

=29k2+6k+1+ 3k+2 

=18k2+12k+2+3k+2

=18k2+15k+4

=18k2+15k+3+1

=36k2+5k+1+1

36k2+5k+1+1is one more than a multiple of 3; therefore, it isn’t divisible by 3.​​ 

Let us now consider all even numbers:​​ n=3k-1.

2n2 + n + 1=23k-12+3k-1+1

=23k-12+3k-1+1

=29k2-6k+1+3k

=18k2-12k+2+3k

18k2-9k+2

=3 6k2-3k+2

3 6k2-3k+2 is two more than a multiple of 3; therefore, it isn’t divisible by 3.​​