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√7 is irrational prove it

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√7 is irrational prove it
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Aayush Verma 1 year, 5 months ago

Let us assume that 7​ is rational. Then, there exist co-prime positive integers a and b such that 7​=ba​ ⟹a=b7​ Squaring on both sides, we get a2=7b2 Therefore, a2 is divisible by 7 and hence, ais also divisible by7 so, we can write a=7p, for some integer p. Substituting for a, we get 49p2=7b2⟹b2=7p2. This means, b2 is also divisible by 7 and so, b is also divisible by 7. Therefore, a and b have at least one common factor, i.e., 7. But, this contradicts the fact that a and bare co-prime. Thus, our supposition is wrong. Hence, 7​ is irrational.
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