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[parent] linear congruence (Definition)

The linear congruence

$\displaystyle ax\equiv b \pmod{m},$
where $ a$, $ b$ and $ m$ are known integers and $ \gcd{(a,\,m)} = 1$, has exactly one solution $ x$ in $ \mathbb{Z}$, when numbers congruent to each other are not regarded as different. The solution can be obtained as
$\displaystyle x = a^{\varphi(m)-1}b,$
where $ \varphi$ means Euler's phi-function.

Solving the linear congruence also gives the solution of the Diophantine equation

$\displaystyle ax\!-\!my = b,$
and conversely. If $ x = x_0$, $ y = y_0$ is a solution of this equation, then the general solution is
\begin{displaymath}\begin{cases} x = x_0\!+\!km,\ y = y_0\!+\!ka,\ \end{cases}\end{displaymath}
where $ k = 0$, $ \pm1$, $ \pm2$, ...



"linear congruence" is owned by Mathprof. [ owner history (1) ]
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See Also: quadratic congruence, solving linear Diophantine equation

Other names:  first degree congruence

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Cross-references: general solution, equation, Diophantine equation, Euler's, congruent, numbers, solution, integers
There are 2 references to this entry.

This is version 11 of linear congruence, born on 2004-04-15, modified 2006-10-02.
Object id is 5763, canonical name is LinearCongruence.
Accessed 3438 times total.

Classification:
AMS MSC: 11A41 (Number theory :: Elementary number theory :: Primes)
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