Primes and Squares again
Q: Show that if a, b are positive integers such at (a,b)=1 and ab is a square, then a and b are also squares.
Pf: (by Contradiction)
Suppose a and b are not squares, then a=h*k and b=p*q where h,k,p,q are all unique primes. (Is this the right approach to this proof?)
Hint: A number is a perfect square iff all exponents in its prime decomposition are even.
Use this on ab. Since (a,b)=1, it is clear how prime decompositions of a and of b must look like.
Could you explain this to me?
Originally Posted by aman_cc
So which means there is an integer h such that
So the power of the prime is 2 but what about h?
I think there's a bit of a problem with your logic that if then all the powers in the prime factorization of will be even. If you think about it , and is not even ...
But if you use Taluivren's hint, you can take the prime decomposition of , and all the powers have to be even. Since , we can rearrange the prime decomposition of so it is written as the prime decomposition of multiplied by the prime decomposition of . Then, we look at the respective prime decompositions and notice that the powers are all even. Again, by the hint, we have that and are also perfect squares.
@Bingk - I get what you are saying. Yes - it is incorrect. Thanks for pointing that out.
Originally Posted by Bingk