Problem 1019

## Preliminary Considerations:

Definition of Prime Number: A prime number is a number that is only divisible by 1 and itself.

Divisibility Theorem: If a number is divisible by 2 and by 3, then it is also divisible by 6.

Therefore, one must prove that either $p-1$ or $p+1$ is divisible by 2 and by 3 to prove that either $p-1$ or $p+1$ is divisible by 6.

Divisibility by 2:

$p$ is a prime number and greater than 3, so it cannot be divisible by 2, as a prime number is only divisible by 1 and itself. Since every second number is divisible by 2, the neighboring numbers of $p$ ($p-1$ and $p+1$) must both be divisible by 2.

Divisibility by 3:

$p$ cannot be divisible by 3 according to the definition of prime numbers (see above), since $p$ is a prime number. If we assume that $p-1$ is not divisible by 3, then $p+1$ must be divisible by 3, as every third number is divisible by 3. This also applies to $p-1$ if $p+1$ is not divisible by 3. Therefore, either $p-1$ or $p+1$ is divisible by 3.

$\Rightarrow$ Since $p-1$ and $p+1$ are divisible by 2 and either $p-1$ or $p+1$ is divisible by 3, either $p-1$ or $p+1$ is divisible by 6.