Residue Class
المؤلف:
Shanks, D
المصدر:
Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea
الجزء والصفحة:
...
14-1-2020
1347
Residue Class
The residue classes of a function 
mod 
are all possible values of the residue 
. For example, the residue classes of 
(mod 6) are ![<span style=]()
{0,1,3,4}" src="http://mathworld.wolfram.com/images/equations/ResidueClass/Inline5.gif" style="height:15px; width:62px" />, since
are all the possible residues.
A complete residue system is a set of integers containing one element from each class, so ![<span style=]()
{0,1,9,16}" src="http://mathworld.wolfram.com/images/equations/ResidueClass/Inline6.gif" style="height:15px; width:69px" /> would be a complete residue system for 
(mod 6).
The 
residue classes prime to 
form a group under the binary multiplication operation (mod 
), where 
is the totient function (Shanks 1993) and the group is classed a modulo multiplication group.
REFERENCES:
Nagell, T. "Residue Classes and Residue Systems." §20 in Introduction to Number Theory. New York: Wiley, pp. 69-71, 1951.
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 56 and 59-63, 1993.
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