Let $a,b,c>0$ be natural numbers. Consider the following statments:
i) if $a\nmid b$ and $b |c$ then $a\nmid c$
ii) if $a |b$ and $b |c$ then $ab |bc$
iii) if $a |c$ and $b |c$ then $ab |c$
iv) If $a |b$ and $b |c$ and $c |a$ then $ac |b^2$
Question: Determine whether each statement is true or false.
$q_1,q_2,q_3$ are natural numbers
So for i) a is not a factor of b, and b divides c, say $c=q_1b$ so in the case when a is a factor of $q_1$ this is false.
for ii) a divide b implies $a |b=aq_1$ and b divides c so $aq_1 |c=aq_1q_2$ and as $aaq_1 |aq_1aq_1q_2$ which is true..
for iii) a divides c implies $a |c=aq_1$, b divides c implies $b |c=bq_2$ so ab does not divide c when a is not a factor of $q_2$ or b is not a factor of $q_1$ so false
for iv) a divides b so $a |b=aq_1$ b divides c $b=aq_1 |c=aq_1q_2$ and c divides a $aq_1q_2 |a$ which implies $q_1,q_2$ are 1 so this means that a,b and c must beequal so this is always true.
This seems like a really long way to do this is it right and is there a nicer way to get this done?