# Homework Help: Abstract Algebra proof

1. Jul 26, 2010

### stihl29

Let a, b be integers a,b>0 show that if a^3 | b^2 then a|b
(Consider the prime factorization of a and b)

I've tried setting up generic prime factorization of a and b but then dont get any where, i'm not very strong at this subject.

Any kind of hints / where to start would help alot thanks!

2. Jul 26, 2010

### Dick

You already have a perfectly good clue. How did you try to use prime factorization? Show the number of times any prime p divides a is less than or equal to the number of times p divides b.

Last edited: Jul 26, 2010
3. Jul 26, 2010

### stihl29

two things, how should i show a prime p, divides a ex.
(p=2^e1 *3^e2 *5^e3....) = a*q?, q is an integer

And why do i need to show that it is less than or equal to the number of times p divides b?

4. Jul 26, 2010

### Dick

Let's do the second one first. The only way a can divide b is if the number of times every prime p divides a is less or equal to the number of times p divides b. Think about the prime factorization of b/a. Don't you agree?

5. Jul 26, 2010

### stihl29

yes i agree with what you are saying here.

6. Jul 26, 2010

### Dick

Well, ok. So then if the largest power of p in a is p^ka and the largest power of p in b is p^kb, what must be true if a^3 divides b^2?

7. Jul 26, 2010

### stihl29

do you mean a^3 divides b^2? if so then, b would be larger than a?

8. Jul 26, 2010

### Dick

No, I'm asking you to compare the number of times p divides a^3 versus the number of times p divides b^2. The first must be less than or equal to the second, right? What does that tell you about ka and kb?

9. Jul 26, 2010

### stihl29

kb must be bigger than ka?

10. Jul 26, 2010

### Dick

Yes. Why does a^3 divides b^2 tell you that? Please help me here. I can't just tell you what to write down. You have to understand it.

11. Jul 26, 2010

### stihl29

is it that there is some factor times b that makes a=b?

12. Jul 26, 2010

### rs1n

The best way to start is to just try some simple examples. Make up some small examples which you know will work / not work.

For example, see what happens for a=15 and b=225 (use prime factorization, as hinted).

Now try something like a=15 and b=75. Why doesn't this example work?

See if you can then come up with the general idea.