Prooving a statement with a Lemma

alexk307

1. The problem statement, all variables and given/known data
Using the lemma below, prove that if two integers divide each other, then they are equal

Lemma: If the product of two integers is 1, then the integers each equal 1.


2. Relevant equations



3. The attempt at a solution
Very lost here, I can format the proof but I don't know where to start it. Also, isn't the lemma false? If a*b = 1, then a and b equal 1. What about 3/4 * 4/3 = 1? or 2/3 * 3/2 = 1?

gb7nash

Ah, but you're forgetting the hypothesis of the lemma. What's the hypothesis? The hypothesis is assumed to be true so....

Once you understand the lemma, what does it mean for one integer to divide another integer?

Mark44

3/4 and 4/3 aren't integers, nor are 2/3 and 3/2.

alexk307

Oh man, that's embarrassing... Thanks

alexk307

okay so,
I assumed that a and b are both integers and that a/b = 1.
Should I also assume that c*d=1 according to the lemma that if cd=1 then c and d are 1?

gb7nash

Start with the hypothesis of what you're trying to prove. Using math terms, what does it mean if a divides b? b divides a?

alexk307

Okay, so if a divides b, then a = bq, where q is the multiplier, also there is no remainder.

so I see that a = bq and b=aq

solving for q... q^2=1 and q=1

I don't know if I'm onto anything by saying q*q=1, and the lemma are related.

alexk307

just got it I think. because q*q = 1, q must equal 1. Therefor a = bq and b = aq can be reduced to a=b and b=a.

gb7nash

Not quite. It's true that a = bq, where q is an integer. However, you cannot say b = aq. It could be a different integer, so the simple solution is to make b = ar, where r is an integer.

So now that you have the two equations:

a = bq
b = ar

What can we do?

alexk307

oh okay so a=bq and b=ar solving for then rq=1.

So I had the right idea, but I can't say that they're both able to be divided by the same integer, but that rq=1

and then r and q both equal 1, so a=b and b=a.

gb7nash

You got it.

alexk307

very helpful, thanks for the walkthrough!