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
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This note was uploaded on 11/03/2010 for the course MATH 541 taught by Professor Pollack during the Fall '09 term at BU.

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

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