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HW4_Solutions

HW4_Solutions - Math 541 Solutions to HW#4 1 Use the...

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Math 541 Solutions to HW #4 1. Use the Euclidean algorithm to compute the greatest common divisor of 320353 and 257642. 320353 = 257642(1) + 62711 257642 = 62711(4) + 6798 62711 = 6798(9) + 1529 6798 = 1529(4) + 682 1529 = 682(2) + 165 682 = 165(4) + 22 165 = 22(7) + 11 22 = 11(2) + 0 We conclude that g.c.d.(320353, 257642) = 11. 2. Prove that the equation 320353 x + 257642 y = 1 has no solutions with x, y in Z . (Please do this question directly without referring to theorems from class.) From (1), we know that g.c.d(320353, 257642) = 11. We can use this to determine that 320353 = 11 · 29123, and 257642 = 11 · 23422. Then 320353 x + 257642 y = (11 · 29123) x + (11 · 23422) y = 11 · (29123 x +23422 y ). That is, 11 divides any integral solution of the equation 320353 x +257642 y . Since 11 does not divide 1, we conclude that the equation has no integer solutions. 3. Compute the following values: φ (100), φ (40), φ (101). [Recall that we have formulas φ ( p n ) = p n - p n - 1 and φ ( mn ) = φ ( m ) φ ( n ) if m and n are relatively prime.] φ (100) = φ (2 2 · 5 2 ) = (2 - 1)(2 1 )(5 - 1)(5 1 ) = (1)(2)(4)(5) = 20 φ (40) = φ (2 3 · 5 1 ) = (2 - 1)(2 2 )(5 - 1)(5 0 ) = (1)(4)(4)(1) = 16 φ (101) = (101 - 1)(101 0 ) = 100 (101 is a prime!) 4. Give 5 examples of groups with 8 elements. Do these groups have distinct multiplication tables up to reordering? ( Z 8 , +), i.e. Z 8 under addition U (15), since φ (15) = φ (3 · 5) = (3 - 1)(5 - 1) = (2)(4) = 8. U (16), since φ (16) = φ (2 4 ) = (2 - 1)(2 3 ) = (1)(8) = 8. U (20), since φ (20) = φ (2 2 · 5) = (2 - 1)(2 1 )(5 - 1) = (1)(2)(4) = 8. U (24), since φ (24) = φ (2 3 · 3) = (2 - 1)(2 2 )(3 - 1) = (1)(4)(2) = 8. The multiplication tables for Z 8 , U (15) , U (16) , U (20) , U (24) follow: 1

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( Z 8 , +) = + 0 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 0 2 2 3 4 5 6 7 0 1 3 3 4 5 6 7 0 1 2 4 4 5 6 7 0 1 2 3 5 5 6 7 0 1 2 3 4 6 6 7 0 1 2 3 4 5 7 7 0 1 2 3 4 5 6 U (15) = · 1 2 4 7 8 11 13 14 1 1 2 4 7 8
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HW4_Solutions - Math 541 Solutions to HW#4 1 Use the...

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