sol1 - Problem Set 1, 18.100B/C, Fall 2011 Michael Andrews...

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Unformatted text preview: Problem Set 1, 18.100B/C, Fall 2011 Michael Andrews Department of Mathematics MIT September 21, 2011 1 Let m and n be positive integers with no common factor. Prove that if p m/n is rational, then m and n are both perfect squares, that is to say there exist integers p and q such that m = p 2 and n = q 2 . (This is proved in Proposition 9 of Book X of Euclids Elements). Assume p m/n is rational. Then there exist positive integers M and N with no common fac- tor such that p m/n = M/N and so mN 2 = nM 2 . Claim: M 2 divides m and N 2 divides n . Assume the claim for now. Then m = M 2 m and n = N 2 n for some m and n . Substituting we obtain M 2 m N 2 = N 2 n M 2 which gives m = n . m = n divides m and n so m = n = 1 and we have shown m and n are perfect squares. Proof of claim: We show that M 2 divides m ; the argument that N 2 divides n is identical. Write M as a product of primes p 1 p r and note that no p i divides N . Assume inductively that p 2 1 p 2 t divides m . Then p 2 t +1 M 2 p 2 1 p 2 t m p 2 1 p 2 t N 2 Since p t +1 does not divide N 2 we see p 2 t +1 m p 2 1 p 2 t , which gives p 2 1 p 2 t +1 m. The inductive hypothesis holds when t = 0; the empty product is 1. Thus, by induction p 2 1 p 2 r = M 2 divides m . 1 2 Problem 6 from page 22. Fix b > 1 a m n = p q = mq = pn so that (( b m ) 1 /n ) nq = ( b m ) q = b mq = b pn = ( b p ) n = (( b p ) 1 /q ) nq . By uniqueness of nq th roots ( b m ) 1 /n = ( b p ) 1 /q . b Let r,s Q . Write r = m/n and s = p/q where m,p Z and n,q N . Since nq is an integer we know that ( b r b s ) nq = ( b r ) nq ( b s ) nq but ( b r ) nq = (( b m ) 1 /n ) nq = ( b m ) q = b mq and similarly ( b s ) nq = b np . Since mq and np are integers we can conclude ( b r b s ) nq = b mq b np = b mq + np Since there is a unique positive real number y such that y nq = b mq + np , we obtain b r b s = ( b mq + np ) 1 /nq = b mq + np nq = b r...
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This note was uploaded on 02/15/2012 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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sol1 - Problem Set 1, 18.100B/C, Fall 2011 Michael Andrews...

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