Prove that root 6 is an irrational number

1.	Prove that root 6 is an irrational number
Answer» We have to prove that {tex} \\sqrt6{/tex}\xa0is an irrational number.Let\xa0{tex}\\sqrt6{/tex}\xa0be a rational number.{tex}\\therefore \\quad \\sqrt { 6 } = \\frac { p } { q }{/tex}where p and q are co-prime integers and\xa0{tex}q \\neq 0{/tex}On squaring both the sides, we get,or,\xa0{tex}6 = \\frac { p ^ { 2 } } { q ^ { 2 } }{/tex}or, p2 = 6q2{tex}\\therefore{/tex}\xa0p2 is divisible\xa0by 6.p is divisible by 6........(i)Let p = 6r for some integer ror, p2\xa0= 36r26q2= 362 [∵ p2 = 6q2]or, q2 = 6r2or, q2\xa0is divisible by 6.{tex}\\therefore{/tex}q is divisible by 6..........(ii)From (i) and (ii)p and q are divisible by 6, which contradicts the fact that p and q are co-primes.Hence, our assumption is wrong.{tex}\\therefore{/tex}\xa0{tex}\\sqrt6{/tex}\xa0is irrational number.

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