Sum and product of coprime numbers

Question

Lets suppose we have $a$ and $b$ which are natural numbers and coprime.

Are $a+b$ and $a\cdot b$ then coprime to each other?

I can't find any information on that on the internet so I'd be thankful if anyone could answer my question.

Thanks a lot in advance.

Answer 1

Hint: Suppose $p$ is a prime dividing both $a+b$ and $ab$. Since $p$ divides $ab$, it divides [fill the blanks]. But then since $p$ divides $a+b$, it divides [fill the blanks].

Answer 2

Suppose $a + b$ and $ab$ are not coprime. Let $p$ be a prime such that $p \mid \gcd(a+b,ab)$. Since $p$ is prime, by Euclid's Lemma, $p \mid a$ or $p \mid b$ (assume WLOG that $p \mid a$).

Now $p \mid (a + b)$, so we must have $p \mid (a + b) - a \Rightarrow p \mid b$. This means that $p \mid \gcd(a,b)$, contradicting that they're coprime.