Divisibility on product of two numbers

Question

Let $p$ be a prime, and $a$ and $b$ two integers, such that $p\not|$ $a$ and $p\not|$ $b$.

It can be possible that $p$ $|$ $ab$ ?

Answer 1

Hint. Recall that a prime $p$ divides an integer $a$ iff $p$ is in the integer factorization (which is unique) of $a$. If you know the integer factorizations of $a$ and $b$, what is the integer factorization of their product? What may we conclude?

Answer 2

Hint: The contrapositive of

$(p|ab) \Rightarrow (p|a$ or $p|b)$

answers your question.

Answer 3

No, it is not possible. If $p\not|a$ and $p\not|b$ then $p$ is not in prime factorisation of $a$ and $b$. Hence, $p$ is not in prime factorisation of $ab$

Answer 4

In $\mathbb{N}$ we have a unique factorization, it means $a=p_1^{n_1}...p_k^{n_k}$. So if $p$ doesn't appear in factorization of a or b, it means it doesn't appear in $ab$.