# modular arithmetic

(redirected from Congruence class)
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Related to Congruence class: Modulo arithmetic

## modular arithmetic

(mathematics)
(Or "clock arithmetic") A kind of integer arithmetic that reduces all numbers to one of a fixed set [0..N-1] (this would be "modulo N arithmetic") by effectively repeatedly adding or subtracting N (the "modulus") until the result is within this range.

The original mathematical usage considers only __equivalence__ modulo N. The numbers being compared can take any values, what matters is whether they differ by a multiple of N. Computing usage however, considers modulo to be an operator that returns the remainder after integer division of its first argument by its second.

Ordinary "clock arithmetic" is like modular arithmetic except that the range is [1..12] whereas modulo 12 would be [0..11].
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Let A mod M be a congruence class containing a squarefree integer, and suppose that A mod M is not entirely contained in the residue class 7 mod 8.
Then [a]D = [b]D and so by the property of congruence class, we obtain a * b [member of] D and b * a [member of] D.
We next describe the connection between the special subsets and the congruence classes of a BCK-algebra.
For x [member of] [S.sub.n], the congruence class of x mod [THETA] is denoted by [[x].sub.[THETA]].
We define a map r: K[[S.sub.[infinity]]] [right arrow] K[[Av.sub.[infinity]]] that identifies the representative of a congruence class. Given
We prove the existence of a unique factorization of a contextual trace as a product of images, in the canonical morphism [phi], of words related to the Lukasiewicz words and characterize the lexicographically minimum and maximum representatives of a congruence class. This allows us to give a closed formula expressing the number of elements of T of a given length.
If w is a representative of the c-trace x [member of] T, then its length and its excess are invariants of its congruence class, so we may use the notations [absolute value of x] and exc(x).
We also investigate the complete semilattice congruence classes of S.
If the group of left-regular permutations of the quasigroup (K, *) is trivial,then the orbits of the group of left-regular permutations are contained in the congruence classes of the extension.
Since the intrinsic volumes are motion invariant characteristics, their weights can be presented in terms of the congruence classes instead for the complete set of pixel configurations.
We found some conditions necessary for a collection of congruence classes to cover the integers.
It is easy to verify that congruence classes are always intervals in L.

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