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In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the...

{\displaystyle {\begin{aligned}2&\equiv -3{\pmod {5}}\\-8&\equiv {\phantom {+}}7{\pmod {5}}\\-3&\equiv -8{\pmod {5}}.\end{aligned}}} The congruence relation...

{\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}} Zero...

non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can...

prime field of order p {\displaystyle p} may be constructed as the integers modulo p {\displaystyle p} , Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} }...

{\displaystyle B\,{\widehat {=}}\,1} , etc.), and addition is performed modulo 26, Vigenère encryption E {\displaystyle E} using the key K {\displaystyle...

{\begin{aligned}s&=(0\times 10)+(3\times 9)+(0\times 8)+(6\times 7)+(4\times 6)+(0\times 5)+(6\times 4)+(1\times 3)+(5\times 2)\\&=130.\end{aligned}}} Adding...

quadratic residue modulo p {\displaystyle p} if it is congruent to a perfect square modulo p {\displaystyle p} and is a quadratic nonresidue modulo p {\displaystyle...

\end{aligned}}} Constructing the sum not over all primes but only over primes which are in the same modulo class introduces further...

{\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}...

a quadratic residue modulo n. This is because for a to be a quadratic residue modulo n, it has to be a quadratic residue modulo every prime factor of...

a prime, then the only square roots of 1 modulo n are 1 and −1. Proof Certainly 1 and −1, when squared modulo n, always yield 1. It remains to show that...

{\begin{aligned}\sum _{n=0}^{\infty }f_{2n}z^{2n}&={\frac {F(z)+F(-z)}{2}}\\[4px]\sum _{n=0}^{\infty }f_{2n+1}z^{2n+1}&={\frac {F(z)-F(-z)}{2}}.\end{aligned}}}...

arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most...

{\displaystyle {\begin{aligned}&0\cdot 8+3\cdot 7+7\cdot 6+8\cdot 5+5\cdot 4+9\cdot 3+5\cdot 2\\&=0+21+42+40+20+27+10\\&=160\;.\end{aligned}}} The remainder...

S p a c i n g = ( c P R F ) {\displaystyle {\begin{aligned}Transmitter\\Characteristics\end{aligned}}{\begin{cases}Duty\ Cycle=\mathrm {PRF} \times Transmit\...

again modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure...

congruential generator (LCG) that operates in multiplicative group of integers modulo n. The general formula is X k + 1 = a ⋅ X k mod m , {\displaystyle X_{k+1}=a\cdot...

is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean...

some of these issues are addressed in Excel 2010. Excel has issues with modulo operations. In the case of excessively large results, Excel will return...