HW 8.docx - Homework Assignment 8 Chapter 21 2 Given p=7...

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Homework Assignment 8 Chapter 21 2. Given: p=7 1<a<7 a m ≡1(mod7) Using the derivation of Fermat’s theorem: 1 : 2 3 3 6 4 6 5 6 6 2 m p when a m a m a m a m a m     6. 10. Given: 13 2 =169 Thus: 2 2 2 2 17 17 17 13 0 13 norm a b a b if a b then b therefore a 13 is not an integer
16. Given: 6 5 4 3 2 1 0 x x x x x x Solve using Gauss’s method: 1 2 6 5 4 3 2 7 6 5 4 3 2 6 5 4 3 2 7 7 1 ( 1)( ... 1) : ( 1)( 1) 0 1 0 1 0 1 p p p x x x x therefore x x x x x x x x x x x x x x x x x x x x x x Let 7 2 1 i x e   : 7 1 7 cos(2 ) sin(2 ) (cos(2 ) sin(2 )) 2 2 cos( ) sin( ) 7 7 x i x i x i In the original equation x 1, therefore: 2 2 cos( ) sin( ) 7 7 n n x i only when n=1,2,3… 32. If AB=0 And * AB A B Then, the determinant of either A or B must be 0 For example:
( )( ) ( )( ) a b A A ad cb c d e f B B eh gf g h ae bg af bh AB ce dg cf dh AB ae bg cf dh ce dg af bh If A ad cb = 0 or B eh gf = 0 , then: ( )( ) ( )( ) AB ae bg cf dh ce dg af bh = 0 38. Given: 2 2 3 0 5 3 4 3 6 0 8 12 0 u v x y z u v x y z u v x y z The order of the maximal nonvanishing determinant is 5 variables – 3 equations = 2 arbitrary values Solve explicitly:

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