MHB Prime Versus Irreducible Numbers

[Panda1]

I don't quite know where to start with this one:

"A natural number p>1 is called irreducible if it has the property that, for any natural numbers a and b, p|ab always implies that either p|a or p|b (or both).

Prove that if a natural number p>1 is irreducible, then it also has the property that p=ab always implies that either p=a or p=b."

I figured that I essentially need to prove that irreducible natural numbers greater than 1 are prime.

Any help with this would greatly appreciated especially since I'm rather new to this site.

 

[Euge]

Hi Panda,

Let $p$ be irreducible; let $a, b$ be natural numbers such that $p = ab$. Then $p | ab$, which implies $p | a$ or $p | b$ by irreducibility of $p$. If $p | a$, then since $a | ab = p$ we have $p = \pm a$. Since $p$ and $a$ are positive, $p = a$. Similarly if $p | b$, then since $b | p$ we deduce $p = b$.