Proof: a^3 divides b^2 implies a divides b in Abstract Algebra.

[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 don't get any where, I'm not very strong at this subject.

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

 

[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.

 

[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?

 

[Dick]

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?

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?

 

[stihl29]

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?

yes i agree with what you are saying here.

 

[Dick]

yes i agree with what you are saying here.

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?

 

[stihl29]

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

 

[Dick]

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

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?

 

[stihl29]

kb must be bigger than ka?

 

[Dick]

kb must be bigger than ka?

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.

 

[stihl29]

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

 

[rs1n]

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

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.