Given below is the proof that if √2 is irrational. Which of the following is the first incorrect step?
STEP-1: Let us assume, to the contrary, that √2 is rational. STEP-2: So, we can find integers r and s (≠0) such that √2=r/s STEP-3: Suppose r and s have a common factor other than 1 STEP-4: √2=a/b (dividing by the common factor, where a and b are coprime) STEP-5: ⇒ b√2=a STEP-6: ⇒ 2b²=a² (squaring on both sides and rearranging) STEP-7: ∴ 2 divides a² (from STEP-6) STEP-8: ∴ 2 divides a (if 2 divides a², then 2 divides a) STEP-9: ∴ a=2c for some integer c STEP-10: 2b²=4c² [from STEP-6,9] STEP-11: ⇒ b²=2c² STEP-12: ∴ 2 divides b² [from STEP-11] STEP-13: ∴ 2 divides b (if 2 divides b², 2 divides b) STEP-14: ∴ a and b have at least 2 as a common factor [from STEP-8,12] STEP-15: But this contradicts the fact that a and b have no common factors other than 1 [from STEP-4] STEP-16: This contradiction has arisen because of our incorrect assumption that √2 is rational. STEP-17: ∴ we conclude that √2 is irrational.
(A)
STEP-2
(B)
STEP-8
(C)
STEP-5
(D)
none of given options
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