Question about irrational numbers

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SUMMARY

The discussion centers on proving that \(\sqrt{p/q}\) is an irrational number when \(p\) and \(q\) are distinct primes. The proof begins by assuming \(\sqrt{p/q} = a/b\), where \(a\) and \(b\) are coprime integers. This leads to the equation \(pb^2 = qa^2\). The conclusion is reached through contradiction, demonstrating that if \(b\) were divisible by \(p\), it would violate the condition that \(a\) and \(b\) are coprime.

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olcyr
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Let p and q be distinct primes. Prove that \sqrt{p/q} is a irrational number.
 
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olcyr said:
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.
 
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)
 
I din't understand why b isn't divisible by p.

Thank you for your answer!
 
because if b is divisible by p, than GCD (a,b) is at least p, but we assumed that it equals to 1
 
Thanks! :)
 

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