Proving Divisibility: Exploring Two Questions on Positive Integers

[1+1=1]

Two questions here. I know the definitions, but cannot formulate a through proof.

1.a and b are positive integers. If a^3 | (is divisible by) b^2, then a | (is divisible by) b.

Now, by definition, I know that a^3*k=b^2, for some k. Also, I know that a * j = b for some j. But where do I go from here?

2.If p^a || (exactly divides) m, then p^ka || (exactly divides) m^k.

Again, by definition, p^a | (is divisible by) m and p^a+1 is not divisible by m. Also, p^ka | (is divisible by) m^k and p^ka+1 is not divisible by m^k+1.

This is all I can get. I just do not know where to go from here. Does anyone have any suggestions?? Thank you all, and you are all very smart on this website, if I have never mentioned that before!

 

[EnumaElish]

a^3 k = b^2
a (a^2 k) = b b
a (a^2 k/b) = b
j = a^2 k/b

Show j is the "same kind of number" as k. If k is a positive integer then show that so is j.

 

[1+1=1]

Ahhhh, I knew it was something with algebra. Thank you very much. I understand that now. Becuase something multiplied by a must mean that a is divisible by b. Thanks much! If only I can get this second one.

 

[Gokul43201]

Two questions here. I know the definitions, but cannot formulate a through proof.
1.a and b are positive integers. If a^3 | (is divisible by) b^2, then a | (is divisible by) b.
Now, by definition, I know that a^3*k=b^2, for some k. Also, I know that a * j = b for some j.

It's the other way round : a|b means that a divides b, or b is divisible by a.

 

[Gokul43201]

a^3 k = b^2
a (a^2 k) = b b
a (a^2 k/b) = b
j = a^2 k/b

Show j is the "same kind of number" as k. If k is a positive integer then show that so is j.

I don't see how this has reduced the difficulty of the problem ... ..or what the OP has understood from it.

 

[matt grime]

1. One only ever need consider prime factors, with multiplicty, for a direct proof. Or you could prove it by contradiction.

2. is easier. p^a exactly divides n is the same as saying n=m*p^a where p does not divide m. The solution should just leap out at you.