# Hw3 - p90 E8(i Empty set p90 E12(i If c and d are inverses of a and b respectively then cd ab = c a d b =[1(ii If ba = ba then by multiplying both

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MATH 3360 HOMEWORK SOLUTION 3 (Some computational problems are omitted.) p72, E2 1729 = 7 × 13 × 19. Hint: 3 6 1( mod 7), 3 12 1( mod 13) and 3 18 1( mod 19). p72, E3 0 , 9 , 18 , 27 p72, E7 We know r ( a - b ) = km , then divide both sides by r , we get: a - b = km r . If you factor out the gcd of m and r , it becomes: a - b = k r/ ( r,m ) m ( r,m ) k r/ ( r,m ) must be an integer because r ( r,m ) is relatively prime to m ( r,m ) . p73, E2 All are possible except 24. p74, E8 Find as + mt = 1, then multiply both sides by r - b . p86, E8 (a) Use page 72 E7. (b) Suppose ( b,m ) = r > 1. Then we claim that it cannot happen that ba i 1( modm ), as r divides both ba i and m but doesn’t divide 1. So { ba 1 ,ba 2 ,...,ba m } is not a complete list. p89, E5 [3][7] = [1], [9][9] = [1], [11][11] = [1], [13][17] = [1], [19][19] = [1].
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Unformatted text preview: p90, E8(i) Empty set. p90, E12 (i) If [ c ] and [ d ] are inverses of [ a ] and [ b ], respectively, then [ cd ][ ab ] = [ c ][ a ][ d ][ b ] = [1]. (ii) If [ ba ] = [ ba ], then by multiplying both sides by the inverse of [ b ] one can get [ a ] = [ a ], contra-diction. (iii) First of all, each [ ba i ] is a unit by (i), and they are diﬀerent by (ii). The number of units is a ﬁxed number for each m , so [ ba 1 ] ,..., [ ba r ] is a complete list of units. 1...
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## This note was uploaded on 09/05/2011 for the course MATH 3360 taught by Professor Billera during the Spring '08 term at Cornell University (Engineering School).

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