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# sep13 - a-b = 1 mod 11 and the only possibility is to...

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Math 3124 Tuesday, September 13 September 13 Ungraded Homework Problem 11.14 page 65 Prove or disprove that Z # 4 is a group with respect to . Z # 4 is not a group under . The reason is that closure for multiplication fails, so is not an operation. This is because [ 2 ] [ 2 ] = [ 4 ] = [ 0 ] . Problem 11.15 page 65 Prove that { [ 0 ] , [ 2 ] , [ 4 ] } is a subgroup of Z 6 . Construct the Cayley table for the subgroup. It follows from the Cayley table that { [ 0 ] , [ 2 ] , [ 4 ] } is a subgroup of Z 6 (the operation is assumed to be ). [0] [2] [4] [0] [0] [2] [4] [2] [2] [4] [0] [4] [4] [0] [2] The invalid ISBN 2406214061 is the result of having two adjacent digits not involving the check digit being transposed. Determine the correct ISBN. Let the n th and ( n + 1 ) st digits be a and b respectively. If we interchange these two digits, the sum we have to calculate is changed from ( 11 - n ) a +( 11 - n - 1 ) b to ( 11 - n ) b +( 11 - n - 1 ) a , which is an increase of b - a . In other words, we want to find two adjacent digits a , b such that when we increase the relevant sum by a - b , it becomes divisible by 11. The relevant sum is 10 * 2 + 9 * 4 + 8 * 0 + 7 * 6 + 6 * 2 + 5 * 1 + 4

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Unformatted text preview: a-b = 1 mod 11, and the only possibility is to interchange the 2 and 1. Thus the correct ISBN is 2406124061 Multiplication of matrices over Z 6 . [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 3 ] [ 1 ] ± [ 1 ] [ ] [ 5 ] [ 5 ] [ 1 ] [ ] ² = [ 1 ] ± [ 1 ] ⊕ [ 2 ] ± [ 5 ] [ 1 ] ± [ ] ⊕ [ 2 ] ± [ 1 ] [ 1 ] ± [ 5 ] ⊕ [ 2 ] ± [ ] [ 3 ] ± [ 1 ] ⊕ [ 4 ] ± [ 5 ] [ 3 ] ± [ ] ⊕ [ 4 ] ± [ 1 ] [ 3 ] ± [ 5 ] ⊕ [ 4 ] ± [ ] [ 3 ] ± [ 1 ] ⊕ [ 1 ] ± [ 5 ] [ 3 ] ± [ ] ⊕ [ 1 ] ± [ 1 ] [ 3 ] ± [ 5 ] ⊕ [ 1 ] ± [ ] = [ 5 ] [ 2 ] [ 5 ] [ 5 ] [ 4 ] [ 3 ] [ 2 ] [ 1 ] [ 3 ] Cayley table for a = ± [ 1 ] [ ] [ ] [ 1 ] ² , b = ± [ 2 ] [ ] [ ] [ 2 ] ² , c = ± [ ] [ 1 ] [ 2 ] [ ] ² , d = ± [ ] [ 2 ] [ 1 ] [ ] ² under the operation of matrix modular multiplication. a b c d a a b c d b b a d c c c d b a d d c a b...
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sep13 - a-b = 1 mod 11 and the only possibility is to...

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