In this question, $a , b$, and $c$ are positive integers.
The following is an attempted proof of the false statement:
If $a$ divides $b c$, then $a$ divides $b$ or $a$ divides $c$.
['$a$ divides $b c$' means '$a$ is a factor of $b c$']
Which line contains the error in this proof?
1. The statement is equivalent to if $a$ does not divide $b$ and $a$ does not divide $c$ then $a$ does not divide $b c$'.
2. Suppose $a$ does not divide $b$ and $a$ does not divide $c$. Then the remainder when dividing $b$ by $a$ is $r$, where $0 < r < a$, and the remainder when dividing $c$ by $a$ is $s$, where $0 < s < a$.
3. So $b = a x + r$ and $c = a y + s$ for some integers $x$ and $y$.
4. Thus $b c = a ( a x y + x s + y r ) + r s$.
5. So the remainder when dividing $b c$ by $a$ is $r s$.
6. Since $r > 0$ and $s > 0$, it follows that $r s > 0$.
7. Hence $a$ does not divide $b c$.