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6 votes
Determine if \sqrt{41}

41

is rational or irrational and give a reason for your answer.

User Kevindra
by
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1 Answer

2 votes

Assume √41 is rational, so that there exist integers p, q such that


√(41) = \frac pq

and p/q is irreducible. Taking squares, this would mean


41 = (p^2)/(q^2) \implies p^2 = 41q^2

This tells us that 41 is a factor of p², which in turn means 41 is a factor of p because 41 is prime. We can consequently write p = 41n for some integer n. But then


p^2 = (41n)^2=41q^2 \implies 41n^2 = q^2

which says 41 also divides q² and hence divides q as well. In other words, p/q is reducible, so we have a contradiction.

User Sandrine
by
9.3k points

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