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blastoise
#1
Feb13-12, 04:16 PM
P: 22
(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 ab|n. Since ab|n and because a and b must be integers they must both be factors of n. Thus, if a|n or b|n is false then ab will not be a factor of n which means ab∤n.
Thus, ab|n if and only a|n and b|n 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 b|n and a|n

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
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