Question about irrational numbers

[olcyr]

Let p and q be distinct primes. Prove that $$\sqrt{p/q}$$ is a irrational number.

 

[olcyr]

Let p and q be distinct primes. Prove that $$\sqrt{p/q}$$ is a irrational number.

It isn't a homework. I just need to prove it!

Thank you,
Olcyr.

 

[csopi]

It's quite easy. Assume, that $$\sqrt{p/q}=a/b$$, where a and b are relative primes, ie GCD (a,b)=1.

This is equivalent to $$pb^2=qa^2$$. Since p and q are distinct primes, p | a^2 => p | a => The right side is divisible by p^2, and this is a contradiction, because the left side is not (because b is not divisible by p, since GCD (a,b)=1)

 

[olcyr]

I din't understand why b isn't divisible by p.

Thank you for your answer!

 

[csopi]

because if b is divisible by p, than GCD (a,b) is at least p, but we assumed that it equals to 1

 

[olcyr]

Thanks! :)