Adv Alegbra HW Solutions 22

Adv Alegbra HW Solutions 22 - 22 (x) If a and b are natural...

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22 (x) If a and b are natural numbers, there there are natural numbers s and t with gcd ( a , b ) = sa + tb . Solution. False. 1.47 Given integers a and b (possibly negative) with a ±= 0, prove that there exist unique integers q and r with b = qa + r and 0 r < | a | . Solution. We have already proved this when a > 0 and b 0. Assume now that a > 0 and b < 0. Now b > 0, and so there are integers q and r with b = qa + r and 0 r < a ; it follows that b =− qb r .I f r = 0, we are done; if r > 0, then b = ( q 1 ) a + ( a r ) and 0 < a r < a (by Proposition A.2(ii), r < 0 implies a r < a . Now assume that a < 0, so that a > 0 (and so | a |=− a . By what we have proved so far, there are integers q and r with b = q ( a ) + r , where 0 r < a ; that is, b = ( q ) a + r , where 0 r < | a | . 1.48 Prove that 2 is irrational using Proposition 1.14 instead of Euclid s lemma.
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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