Find the error in the following statements and give reasons for your answers.
(a) $3 \mid (−3 × (−5)) \implies 3 = −3$
(b) $6 \not \mid (2 × 5 × 7) \implies \gcd (6, 2) = \gcd (6, 5) = \gcd (6, 7) = 1$
(a) $3 \mid (−3 × (−5))$ simply means that 3 divides 15. It doesn't mean 3 equals any of the given factors of 15.
For example, $4 \mid 2^3$ does not mean $4=2$.
If $x \mid yz$ and $y,z$ are prime, then $x=y$ or $x=z$, by Corollary (2.4).
(b) Aside from the false $\gcd(6,2)=1$, the deeper issue is that $a \not \mid pqr$ does mean $a$ is coprime to all $p,q,r$.
To see this, consider $bc \not \mid bqr$, where $b>1$. Here $\gcd(bc,b) \ne 1$.
On the other hand, if $a \not \mid p_2 p_3 p_4$, where $p_i$ are prime, then we can say $a$ is coprime to all $p_2, p_3, p_4$.