Proof by exhaustion (odd and even cases)

Proof

A-level Maths (9MA0)

ORIGINAL

Split the integers into cases that cover all of them, prove the claim in each case, then write the conclusion.

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Question

Prove that, for every integer

n

n^2 + 2

is not divisible by 4.

Every step worked, with the reasoning.

1
Every integer is either even or odd, so consider $n = 2m$ and $n = 2m + 1$ , where $m$ is an integer.
Proof by exhaustion: two cases that between them cover every integer.
2
If $n = 2m$ : $n^2 + 2 = 4m^2 + 2$
Substitute the even form and expand.
3
$4m^2 + 2$ leaves remainder 2 when divided by 4, so it is not divisible by 4.
$4m^2$ is a multiple of 4; adding 2 gives remainder 2.
4
If $n = 2m + 1$ : $n^2 + 2 = 4m^2 + 4m + 1 + 2 = 4(m^2 + m) + 3$
Expand $(2m + 1)^2$ and take out the factor of 4.
5
$4(m^2 + m) + 3$ leaves remainder 3 when divided by 4, so it is not divisible by 4.
$m^2 + m$ is an integer, so $4(m^2 + m)$ is a multiple of 4.
6
Every integer is even or odd, and in both cases $n^2 + 2$ is not divisible by 4, so $n^2 + 2$ is not divisible by 4 for every integer $n$ .
Finish with a conclusion that refers back to every case and restates the claim.

Answer: Proved: $n^2 + 2$ leaves remainder 2 ( $n$ even) or 3 ( $n$ odd) when divided by 4, so it is never divisible by 4.