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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 * 4 + 3 * 0 + 2 * 6 + 1 * 1 which is 1 mod 11. Therefore we need
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 ] [ ] = [ 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 ] , b = [ 2 ] [ ] [ ] [ 2 ] , c = [ ] [ 1 ] [ 2 ] [ ] , d = [ ] [ 2 ] [ 1 ] [ ] 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...
View Full Document

This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online