Friday, 7 August 2015

proof verification - Show that sqrtq is irrational


Let q be a positive integer such that q2 and such that for any

integers a and b, if q|ab, then q|a or q|b. Show that q
is irrational.




Proof;



Let assume q is a rational number, where n0 and gcd(m,n)=1, meaning q=mnq=m2n2



Since n2, q|m^2 \Rightarrow q|m, so m=qt where t\in \mathbb{Z}




By substitute m=qt in the equation qn^2 = m^2, we get n^2=qt^2.



Since tells us that q|n^2 and t^2|n^2, it contradicts with the assumption \gcd (m,n)=1; therefore, \sqrt{q} is irrational.



I get this proof with the assistant of the course, but is there any flaw or mistake? What are the other methods for proving this statement, can you at least give one different method? And how can I improve this proof?

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