1. if a doesn't divide b and a doesn't divide c , then a doesn't divide b times c.
give a counterexample to show this is a false statement.
2. if a divides c and b divides c sometimes, a times b divides c and sometimes it doesn't
show an example when a times b divides c .
show and example when a times b doesn't divide c.
Usually with this language and nothing else specified, we assume we are working over the integers.
(1) is very simple.. don't know how to give a hint, here's such a counterexample, learn from it
a = 2*3
b = 2*5
c = 3*5
(2) this is confusing. Is the first part supposed to mean
if (a divides c) and (b divides c sometimes)
or
if (a divides c and b divides c) sometimes
?
I'll assume the first. But it can be proven with the stricter requirement "a divides c and b divides c always".
Just use simple examples. For the case ab divides c, let c = ab. For ab does not divide c, let a = b = c > 1.