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Class 10 chapter 1 EXERCISE 1.3 1. Prove that √5 is irrational |
| Answer» Solution - let us assume on the contrary that √5 is a rational number . Then there exist co - prime positive integers a and b such that √5 = a/b 5b^ 2 = a^2 5 | a^2 [ 5 | 5b^2 ] 5 | a. ( see theorem 2) (i)......a = 5c for some positive integer c a^2 = 25c^2 5b^2 = 25c^2 [ a^2 = 5b^2 ]b^2 = 5c^2 5 | b^2 [ 5 | 5c^2 ] 5 | b [ see theorem 2 ] (i) .......From (i)and (i) we find that a and b have at least 5 as a common factor . This contradicts the fact that a and b are co- prime Hence , √5 is an irrational number . | |