Let k and n be positive integers. For m E Z define the formula f : Zn → Zk by f([x]n) = [xm]k for x E Z. (a) Prove that f defines a function if and only if k | mn. (b) Suppose m is a positive integer so that k | mn. Prove that f is a bijection if and only if k = n and (m, n) = 1. %3D
Let k and n be positive integers. For m E Z define the formula f : Zn → Zk by f([x]n) = [xm]k for x E Z. (a) Prove that f defines a function if and only if k | mn. (b) Suppose m is a positive integer so that k | mn. Prove that f is a bijection if and only if k = n and (m, n) = 1. %3D
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 23E: Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove...
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