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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
$\equiv$ modulo $m$ is an equivalence relation. That is, $a\equiv a\bmod m$. If $a\equiv b\bmod m$, then $b\equiv a\bmod m$. If $a\equiv b\bmod m$ and $b\equiv c\bmod m$, then $a\equiv c\bmod m$. Addition and multiplication are well-defined modulo $m$. That is,
28 lut 2018 · We say that $a$ and $b$ are inverses modulo $n$, if $ab \equiv 1 \mod n$, and we might write $b = a^{-1}$. For example $17\cdot 113 = 1921 = 120\cdot 16 +1 \equiv 1 \mod 120$, so $17^{-1} = 113$ modulo $120$.
16 cze 2022 · If you allow the use of the "floor function" $\lfloor x\rfloor$ (the greatest integer $\le x$), there is a simple formula. It is a good way of calculating $a\bmod m$ on a simple calculator. $\endgroup$
congruent modulo m if b−a is divisible by m. In other words, a ≡ b(modm) ⇐⇒ a−b = m·k for some integerk. (1) Note: 1. The notation ?? ≡??(modm) works somewhat in the same way as the familiar ?? =??. 2. a can be congruent to many numbers modulo m as the following example illustrates. Ex. 1 The equation x ≡ 16(mod10)
15 lis 2008 · Namely, if we fix a modulo 3 (say we want a ≡ 1 (mod 3)), then we can find a k to make it so: the equation 1 ≡ 2 + 2k (mod 3) is solvable since 3 is relatively prime to 2 (or equivalently, to 20).
A mod C = R1. B = C * Q2 + R2 where 0 ≤ R2 < C and Q2 is some integer. B mod C = R2. (A + B) = C * (Q1 + Q2) + R1+R2. LHS = (A + B) mod C. LHS = (C * (Q1 + Q2) + R1+ R2) mod C. We can eliminate the multiples of C when we take the mod C.
