Prove or disprove that If $a\mid c$ and $b\mid c$, then $ab \mid c$.

Question

So I am not really sure what to do. I know by the definition of divisibility there must exist some integers $k$ and $l$ such that

$$ c= ak \ \text{ and } \ c=bl $$

But now I am stuck and have no clue where to go from here...

I need to show that $c=ab(\text{some integer})$, for it to be divisible, but I do not see the path to take.

Think about prime factors, as you always should when you have divisibility. Suppose all the numbers are powers of the same prime. What happens? — Ross Millikan, Jul 22 '20 at 03:24
With a statement like this, it's often best to look for a counterexample before you start trying to prove that it is true. Can you think of an integer that is divisible by two integers but not by their product? — Clive Newstead, Jul 22 '20 at 03:24
What two positive integers do you know divide $c$? What if $a$ and $b$ are both the larger of those two numbers? — Eric Towers, Jul 22 '20 at 03:26
if a=2, and b=8, and c=8, then we have that a|c, and b|c, but ab does not divide c. So then this would be a disprove type question? — MathyViking, Jul 22 '20 at 03:29
yes exactly that's a good counterexample (or simply $a=b=c=2$) — peek-a-boo, Jul 22 '20 at 03:29
oh okay, I must have been overthinking it. Thank you for all of your guy's help!! — MathyViking, Jul 22 '20 at 03:30
yeah that a=b=c=2 makes sense too. Because ab will be bigger than c which would not yield in an integer. Makes total sense now. — MathyViking, Jul 22 '20 at 03:31
I feel really silly now for not getting it right away. Thank you everyone again. — MathyViking, Jul 22 '20 at 03:32

score 1 · Answer 1 · answered Jul 22 '20 at 04:05

1

If $a= b = c > 1$ then $a|c, b|c, ab \not\mid c$.

answered Jul 22 '20 at 04:05

marty cohen

1 Answers1