Intro to Abstract Math Question about divison of integers.

blastoise

(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

Norwegian

Statement (1) is false. It becomes true if you add the assumption that gcd(a,b)=1.

blastoise

It's false, because you when you say if and only if it is the same things as

If-And-Only-If Proofs
Often, a statement we need to prove is of the form
\X if and only if Y ." We are then required to do
two things:
1. Prove the if-part: Assume Y and prove X.
2. Prove the only-if-part: Assume X, prove Y .

taken from http://infolab.stanford.edu/~ullman/...es/slides1.pdf

Did 1.

But, number 2 is
Assume n is divisible by b and n is divisible by a if n is divisible by ab


Choose n = 8, b = 2 a = 3
n is divisible by b and n is divisible by a but n is not divisible by ab

so it's false
thx norwegian i see what you mean

Deveno

8 is not divisible by 3.

let's pick a better example, where a and b have "some factor in common".

so suppose a = 6, and b = 15, and n = 30. then a|n (because 30 = 6*5), and b|n (because 30 = 15*2), but it's pretty obvious ab = 90 does NOT divide 30 (for one, it's bigger).

in general, you only know that n is divisible by the least common multiple of a and b. in our example above, lcm(6,15) = 30, and indeed 30 divides 30.