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October 27th, 2017, 09:54 AM  #1 
Newbie Joined: Oct 2017 From: Here Posts: 13 Thanks: 0  Square root of multiplication of 2 primes is irrational
**Before you start reading I must say I don't know how to use math symbols so sorry if it's really messy, and if you have a guide somewhere on how to write maths here I'd appreciate  I'm trying to prove that sqrt(ab) is irrational, if a and b are different primes Using contradiction: Assume sqrt is rational: sqrt(ab) = x/y, where x/y is irreducible fraction then I do power of 2 to both sides to get: ab = (x^2)/(y^2), then: (y^2)*ab=x^2, so ab is divisor of x^2, and thus divisor of x. so there is a k that is k*ab = x, put that back in: (y^2)*ab=(k*ab)^2, divide both sides by (ab): (y^2)=(k^2)*ab, so ab is also divisor of y^2, and also of y! So we get that ab is divisor of x and y, contradiction to the assumption that x/y is irreducible fraction. Q.E.D  Is this OK? 
October 27th, 2017, 10:23 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Why does "ab is divisor of x²" imply that ab is a divisor of x?

October 27th, 2017, 10:29 AM  #3 
Newbie Joined: Oct 2017 From: Here Posts: 13 Thanks: 0 
I was thinking about Euclid's lemma but not sure if it applies here too because this lemma speaks of a single prime number and here we have multiplication of 2 primes. So I guess I am wrong 
October 27th, 2017, 10:41 AM  #4 
Newbie Joined: Oct 2017 From: Here Posts: 13 Thanks: 0 
Sorry, I was using Euclid''s lemma wrong!

October 28th, 2017, 05:29 AM  #5 
Newbie Joined: Oct 2017 From: Here Posts: 13 Thanks: 0 
Anyone? I can't find out

October 28th, 2017, 08:13 AM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,814 Thanks: 1046 Math Focus: Elementary mathematics and beyond 
I've seen proofs that the square root of any nonnegative integer that isn't itself a perfect square is irrational. Unfortunately, I can't recall it but I don't think they were particularly difficult. Can you prove that the square root of 2 is irrational and then extend that to all nonsquare integers?

October 28th, 2017, 10:31 AM  #7 
Newbie Joined: Oct 2017 From: Here Posts: 13 Thanks: 0 
I think I did it finally 

Tags 
irrational, multiplication, primes, root, square 
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