# modular arithmetic

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## 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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'One simple idea is try to solve equations with modular arithmetic,' he says.
Such advanced algorithms comprise of modular arithmetic, modular exponentiation, prime number theory, Chinese Reminder Theorem, and Fermat's Little Theorem and Euler's Theorem.
The following discussion employs modulo residue theory to find tests of divisibilty for even numbers less than 60 and elaborates the use of modular arithmetic from number theory in finding these tests.
Results (see Table 1) indicate that there is no difference in the means for the grades of the real security case analysis ( p-value > [alpha]) but there is a significant difference in the case of the modular arithmetic exam (p-value < [alpha]) for at least one pair of groups.
Moving up a level in difficulty, the Diffie-Hellman-Merkle (DHM) method also uses modular arithmetic but involves discrete logarithms instead of factoring.
Another 23 papers cover arithmetic units, domain specific designs, verification and correctness proofs, modular arithmetic, floating-point error analysis, functional approximation, and arithmetic in cryptography.
(5.) This can also be proved easily using the binomial expansion, without using modular arithmetic.
Similarly, in modular arithmetic what counts is not the numerical valve itself but the remainder after division by the modulus (17 in the example).
The author covers random numbers, arrays, Pythagorean triples, containers, and modular arithmetic. Answers for nearly all the exercises are provided in an appendix.
In modular arithmetic, only remainders left over after division of one whole number by another count.
The mathematics, though esoteric, turns out not to be too difficult to learn and use; readers are expected at the very least however, to be familiar with modular arithmetic, that is, the number systems formed by the remainders of the integers after division by the selected modulus--for example, how you can add four hours to 10:00 and get 2:00.
He pairs music and math concepts such as scales and modular arithmetic, octave identification and equivalence relation, intervals and logarithms, equal temperament and exponents, overtones and integers, tone and trigonometry, and tuning and rationality.

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