My textbook gives the definition of prime numbers as such:
To say that x is a prime number means that $x \in N$ and $x \neq 1$ and $\forall a \in N$, $\forall b \in N$, if $x=ab$, then $a=1$ or $b=1$.
I don't understand this definition of prime numbers.
Let's say I choose $x=10$, which I know not to be a prime number. Then $x=10=10 \times 1$ where I have chosen $a=10$ and $b=1$. Doesn't this fulfill the previous definition but also show that it doesn't hold?
Thank you in advance for helping up to clear any confusion.
Best Answer
I believe you are misunderstanding how quantifiers ($\forall$ "for all" and $\exists$ "there exists") work.
A number $p$ is prime if for any $a, b$ with $ab=p$, we have $a=1$ or $b=1$.
$10$ is not prime because we can take $a=5, b=2$ - that is, there exists some pair $a, b$ with $ab=p$ but $a\not=1\not=b$.
In order for a statement of the form "$\forall a . . .$" to hold, the inner statement has to hold no matter what $a$ you pick: so finding some $a$ for which it works, tells you nothing. Basically, you need to show that no counterexample exists.