Prove that the square root of 5 is irrational. Solution This is one of the first

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This is one of the first proofs you learn when you study realanalysis. Suppose 5 is rational, so suppose 5=p/q where we mayassume p, q are positive integers, and have no common factors (elsewe can remove the common factor from both numerator anddenominator). ---------------(*) Square both sides, so that 5= p2/q2 =>5q2 = p2 . => 5| p2 .Note that we may write such a statement since this is an equationinvolving positive integers. Since 5 is a prime, it follows that5|p2 => 5|p, so we can write p = 5s for some integer s.----------(**) Thus plugging this back we get, 5q2=(5p)2=25p2 => q2=5p2 . By the sameargument as before, it follows that 5|q, so we may write q=5t for some integer t----------------(***) But now note that (**) and (***) imply that 5 is a common factor ofboth p and q and this contradicts (*). Hence our assumption that5 is rational is erroneous, so we are done.

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