In this question, $a,b,$ and $c$ are positive integers.

The following is an attempted proof of the false statement:

If $a$ divides $bc$, then $a$ divides $b$ or $a$ divides $c$.

['$a$ divides $bc$' means '$a$ is a factor of $bc$']

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 $bc$'. 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 $b=ax+r$ and $c=ay+s$ for some integers $x$ and $y$. 4. Thus $bc=a(axy+xs+yr)+rs$. 5. So the remainder when dividing $bc$ by $a$ is $rs$. 6. Since $r>0$ and $s>0$, it follows that $rs>0$. 7. Hence $a$ does not divide $bc$.