Refer to image 22Show that for x and y integers, and ma positive integer x = y mod m implies x = y'mod m.

This is a problem of Number Theory, Modular Arithmatic.

Answered: Show that for x and y integers, and m a…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 5E

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30. Prove that any positive integer is congruent to its units digit modulo .
a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .
25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)
Find the smallest integer in the given set. { and for some in } { and for some in }
Prove that if a+xa+y(modn), then xy(modn).

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Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.