Answered: 3. Congruence Let a, b, c > 0, m 2 2 be… | bartleby

Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...

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1. Div and Mod
(a)
Find the quotient ("div") and the remainder ("mod") when:
(i) 23 is divided by 6
(ii) -42 is divided by 5
(b)
Evaluate these quantities:
(i) –101 mod 13
(ii) 199 mod 19
(c)
to 4 modulo 12.
List two negative integers and two positive integers that are congruent
2. Euclidean Algorithm
Use the Euclidean algorithm to find gcd(12345, 678).
3. Congruence
Let a, b, c> 0, m > 2 be integers. Prove that
a = b (mod m)
implies that
ac = bc (mod mc).
4. Modular Inverse
Compute an inverse of 70 modulo 39 by running the extended Euclidean algorithm
and determining the Bézout coefficients.

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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 .
a. Excluding the identity cipher, how many different translation ciphers are there using an alphabet of n "letters"? b. Excluding the identity cipher, how many different affine ciphers are there using an alphabet of n "letters," where n is a prime?
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.)

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