Let q be a positive integer such that q≥2 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 n≠0 and gcd(m,n)=1, meaning √q=mn⇒q=m2n2
Since n2∤m2, q|m2⇒q|m, so m=qt where t∈Z
By substitute m=qt in the equation qn2=m2, we get n2=qt2.
Since tells us that q|n2 and t2|n2, it contradicts with the assumption gcd(m,n)=1; therefore, √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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