A complete residue system modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set. If a_1, a_2,...,a_m is a complete residue system mo...A complete residue system modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set. If a_1, a_2,...,a_m is a complete residue system modulo m, and if k is a positive integer with (k,m)=1, then ka_1+b, ka_2+b,...,ka_m+b is another complete residue system modulo m for any integer b.
The Euler \phi-function is the map \phi : {\mathbb N } \rightarrow {\mathbb N} defined by \phi(n) = 1 for n=1\text{,} and, for n \gt 1\text{,}\phi(n) is the number of positive...The Euler \phi-function is the map \phi : {\mathbb N } \rightarrow {\mathbb N} defined by \phi(n) = 1 for n=1\text{,} and, for n \gt 1\text{,}\phi(n) is the number of positive integers m with 1 \leq m \lt n and \gcd(m,n) = 1\text{.}
We now present several multiplicative number theoretic functions which will play a crucial role in many number theoretic results. We start by discussing the Euler phi-function which was defined in an ...We now present several multiplicative number theoretic functions which will play a crucial role in many number theoretic results. We start by discussing the Euler phi-function which was defined in an earlier chapter. We then define the sum-of-divisors function and the number-of-divisors function along with their properties.
