Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
If the sequence (p^2=5q^2), (p=5k), (q^2=5k^2) appears in the proof of (\sqrt{5}), what is the next correct statement?After proving both (p) and (q) even in the proof of (\sqrt{2}), how should the final conclusion be written?Which option is a wrong proof method in the proof of (\sqrt{3})?Why is it necessary to assume (p) and (q) coprime while proving (\sqrt{5}) irrational?In the proof of (\sqrt{2}), after getting (p^2=2q^2), which statement is correct but does not yet complete the proof?If in the proof of (\sqrt{3}), both (p) and (q) are found divisible by (3), which statement about (\frac{p}{q}) is correct?Which option gives a correct and complete reasoning in the proof of (\sqrt{5})?Which statement correctly describes the difference between the proofs of (\sqrt{2}) and (\sqrt{3})?In the proof of (\sqrt{5}), (q) is divisible by (5) from (q^2=5k^2). Which condition is necessary for this conclusion?If assuming (\sqrt{2}) rational finally makes (\frac{p}{q}) not remain in lowest form, what is the correct conclusion?Which option shows the common deeper idea in the proofs of (\sqrt{3}) and (\sqrt{5})?Which statement shows a logical weakness in the proof of (\sqrt{2})?In the proof of (\sqrt{5}), after getting (p^2=5q^2), which conclusion cannot be drawn immediately?If in the proof of (\sqrt{3}), both (p) and (q) turn out divisible by (3), which greatest common divisor condition definitely breaks?In the proof of (\sqrt{2}), if (p=2k) and (q=2r), how can (\frac{p}{q}) be reduced?Which option gives a result against (\gcd(p,q)=1) in the proof of (\sqrt{5})?If (\sqrt{3}) is assumed rational and (\frac{p}{q}) is in lowest form, what conclusion follows when both (p) and (q) are found divisible by (3)?Which statement is true but given with a wrong reason in the proof of (\sqrt{2})?Which option correctly identifies the proof of (\sqrt{5})?In the proof of (\sqrt{3}), if (p) is divisible by (3) from (p^2=3q^2), what type of number is (k) in (p=3k)?