Universal and existential statements for prime and composite numbers(proof)

I've a discrete mathematics book that has the following written theorems:

$\displaystyle

n\, is\, prime\, \Leftrightarrow \,\forall\, r,s \in \mathbb{Z}^+,\, if\, n\, = r.s\,\, then\, r = 1\, or\, s = 1

$

My question why is this a universal statement on the right? Can't we just use existential statement like this:

$\displaystyle

n\, is\, prime\, \Leftrightarrow \,\exists\, r,s \in \mathbb{Z}^+,\, if\, n\, = r.s\,\, then\, r = 1\, or\, s = 1

$

Also :

$\displaystyle

n\, is\, composite\, \Leftrightarrow \,\exists\, r,s \in \mathbb{Z}^+ \,,\, if\, n\, = r.s\,\, then\, r \neq 1\, and \, s \neq 1

$

Alternatively can't we write:

$\displaystyle

n\, is\, composite\, \Leftrightarrow \,\forall\, r,s \in \mathbb{Z}^+ \,,\, if\, n\, = r.s\,\, then\, r \neq 1\, and \, s \neq 1

$

Why my alternate method for stating the same theorem not right? Thanks.