(1)Assume a, b and n are nonzero integers. Prove that n is divisible by ab if and
only if n is divisible by a and n is divisible by b.
I'm wrong and can't remember why. I spoke to the professor about it for ~ 1 minute so it seems to have slipped my mind, it was because in one case it's true and in the other it isn't here is my proof:
(2)Let a,b and n be non zero integers and assume abn. Since abn and because a and b must be integers they must both be factors of n. Thus, if an or bn is false then ab will not be a factor of n which means ab∤n.
Thus, abn if and only an and bn where a, b and n are non zero integers.
But, then I pulled from a website "[if and only if ]means you must prove that A and B are true and false at the same time. In other words, you must prove "If A then B" and "If not A then not B". Equivalently, you must prove "If A then B" and "If B then A".
I believe that (2) shows if Statement {A} then {B}.
So how would you show if not Statement {a} then not {B}?
I'm going to say
Suppose ab ∤ n is true then a ∤ n and b∤n
Let a = 10, b = 10, n = 10
ab∤ n, but bn and an
The thing I don't understand is how does that disprove (1).
So, the question I'm asking is: Is statement (1) considered true or considered false taken as is. Also, if you could rip my proof apart would be great help(don't hold back criticize away XD )
Thanks
