(Solved): 6. For all integers \( a, b \), and \( c \) If \( a \mid(b+c) \) then \( a \mid b \) or \( a \mid c ...

6. For all integers \( a, b \), and \( c \) If \( a \mid(b+c) \) then \( a \mid b \) or \( a \mid c \). 7. For all integers \( m \), if 7 is a factor of \( m \) then 7 is not a factor of \( m+6 \). 8. \( (\forall x \in \mathbb{1}) \mid \sqrt{x} \in \mathbb{I}] \) 9. \( (\forall x, y \in \mathbb{Q})(\forall z \in \mathbb{I})[ \) If \( y \neq 0 \) then \( x+y z \in \mathbb{I}] \) 10. \( \log _{5}(2) \in 1 \). Hint: Consider using the Fundamental Theorem of Arithmetic. For each of the following statements, either prove the statement or give a counterexample that shows the statement is false. We will use the (non-standard) notation I to represent the irrational numbers. Each problem is worth 10 points.