\"4\"r) is defined as the element rh such that r.-r-+y(mod m); \"r is that r, fo

Question

"4"r) is defined as the element rh such that r.-r-+y(mod m); "r is that r, for which r , (mod m)]. The reduced residue stem is simply the group of units in this finite ring. Since we are ncerned with the properties of integers rather than with the systems that may be constructed from them, we refer stu- interested in the algebraic approach to Chapter 1 of A Survey gebraic lents f Modern Algebra by G. Birkhoff and S. MacLane. EXERCISES 1. Which of the following are complete residue systems modulo 11? (a) 0, 1,2, 4, 8, 16, 32, 64, 128, 256, 512 (b) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 (c) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 2. Which of the following are reduced residue systems modulo 18 (a) 1, 5, 25, 125, 625, 3125 (b) 5, 11, 17, 23, 29, 35 (c) 1, 25, 49, 121, 169, 289 (d) 1, 5, 7, 11, 13, 17 3. Suppose (a,, a, .. . a) is a complete residue system modulo k, where k is a prime. Prove that for each integer n and each nonnegative integer s there exists a congruence of the form where each b, is one of the a

Explanation / Answer

Hi,
1 A complete residue system modulo n is defined as set of integer which yield all possible remainders i,e from 0 to n-1
given modulo 11, so set should have min 11 elements,
a. 0,1,2,4,8,16,32,64,128,256,512- here all the remainders from 0-10 are covered hence true
b. 1,3,5,7,9,11,13,15,17,19,21- here also all the remainders from 0-10 are covered hence true
c.2,4,6,8,10,12,14,16,18,20,22-  here also all the remainders from 0-10 are covered hence true
d.-5,-4,-3,-2,-1,0,1,2,3,4,5- here all remainders are covered, please note -5 modulo 11 is 6, hence true
2. A reducedresidue system modulo n is defined as set of numbers t from a complete residue system modulo n by removing all integers not relatively prime to n.
we have to see if all the digits in each set is relatively prime to 18, i,e no integer divides them both except 1
a.1,5,25,125,625, - since all numbers are  relatively prime to 18- true.
b.5,11,17,23,29,35- all numbers are prime, and even 35 and 18 dont have a common divisor- hence true
c.1,25,49,121,169,289- all of them are squares of primes, except 289 which is  relatively prime to 18 hence true
d.1,5,7,11,13,17- all are prime and  relatively prime to 18, hence true

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