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/ialc/slides/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 abChoose 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]

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/ialc/slides/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

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.

 

[blue_raver22]

for your question and for sharing your proof and thought process. Let's take a closer look at statement (1) and your proof to see where the issue lies.

Statement (1) is a biconditional statement, which means that it consists of two conditional statements connected by "if and only if". Therefore, in order to prove this statement, we need to show that both the forward and backward implications are true.

Forward implication: If n is divisible by ab, then n is divisible by a and n is divisible by b.

Your proof for this part is correct. You correctly showed that if ab divides n, then a and b must both be factors of n. Therefore, n is divisible by both a and b.

Backward implication: If n is divisible by a and n is divisible by b, then n is divisible by ab.

This is where the issue lies in your proof. You correctly showed that if a and b are both factors of n, then ab must also be a factor of n. However, this does not prove that n is divisible by ab. It only proves that if n is divisible by a and n is divisible by b, then ab must also be a factor of n. This is not the same as saying that n is actually divisible by ab.

To prove the backward implication, you need to show that if a and b are both factors of n, then n is divisible by ab. This can be done using the Fundamental Theorem of Arithmetic, which states that every integer can be uniquely expressed as a product of prime numbers. Since a and b are both integers, they can be expressed as a product of primes (e.g. a = p1 * p2 * ... * pk and b = q1 * q2 * ... * qm). Therefore, n can also be expressed as a product of primes (n = p1 * p2 * ... * pk * q1 * q2 * ... * qm). Since ab = (p1 * p2 * ... * pk) * (q1 * q2 * ... * qm), it follows that ab is also a factor of n.

In summary, statement (1) is true and your proof for the forward implication is correct. However, your proof for the backward implication is incomplete and needs to be revised to show that n is actually divisible by ab. I hope this helps clarify the issue and provides you with a better understanding of abstract math and proofs.