# Homework Help: Divisibility property

1. Jul 31, 2010

### annoymage

1. The problem statement, all variables and given/known data

proof the theorem

if a l b and b l a then a=+-b

2. Relevant equations

3. The attempt at a solution

there exist integer p,q such that ap=b and bq=a, then i've no idea how i can relate it to a=+-b.. clue please T_T

2. Jul 31, 2010

$$ap = b = (\frac{a}{q})$$

Multiply by q and divide be a (since b|a <--> a is not 0), giving us:

$$pq = 1$$

Do you follow?

3. Jul 31, 2010

### annoymage

yea yah, i thought that too, but don't know to continue from there too, owhoho, more clue please, ;P

4. Jul 31, 2010

### annoymage

wait, let me think first

5. Jul 31, 2010

### Staff: Mentor

So b = pa and a = qb, for some integers p and q.
Then b = pa = p(qb) = (pq)b.

What can you say about pq?

6. Jul 31, 2010

Remember, p and q are both nonzero integers.

7. Jul 31, 2010

### annoymage

hmm, so pq=1 , hence, p=1 and q=1, hence a=b and b=a, still cannot get +-b, T_T

8. Jul 31, 2010

### annoymage

WAIIITTTT (-1)(-1) also equal 1, wait wait let me think again

9. Jul 31, 2010

### annoymage

so i get

a=b or (a=-b and b=-a)

=> (a=b or a=-b) and (a=b or b=-a)

(a=b or a=-b) is enough to verify it right?

10. Jul 31, 2010