diff --git a/src/algebra/chinese-remainder-theorem.md b/src/algebra/chinese-remainder-theorem.md index cc198dbda..82f3e5eb8 100644 --- a/src/algebra/chinese-remainder-theorem.md +++ b/src/algebra/chinese-remainder-theorem.md @@ -10,32 +10,32 @@ The Chinese Remainder Theorem (which will be referred to as CRT in the rest of t ## Formulation -Let $p = p_1 \cdot p_2 \cdots p_k$, where $p_i$ are pairwise relatively prime. In addition to $p_i$, we are also given a set of congruence equations +Let $m = m_1 \cdot m_2 \cdots m_k$, where $m_i$ are pairwise coprime. In addition to $m_i$, we are also given a set of congruence equations $$\begin{align} - a &\equiv a_1 \pmod{p_1} \\\\ - a &\equiv a_2 \pmod{p_2} \\\\ + a &\equiv a_1 \pmod{m_1} \\\\ + a &\equiv a_2 \pmod{m_2} \\\\ &\ldots \\\\ - a &\equiv a_k \pmod{p_k} + a &\equiv a_k \pmod{m_k} \end{align}$$ -where $a_i$ are some given constants. The original form of CRT then states that the given set of congruence equations always has *one and exactly one* solution modulo $p$. +where $a_i$ are some given constants. The original form of CRT then states that the given set of congruence equations always has *one and exactly one* solution modulo $m$. ### Corollary A consequence of the CRT is that the equation -$$x \equiv a \pmod{p}$$ +$$x \equiv a \pmod{m}$$ is equivalent to the system of equations $$\begin{align} - x &\equiv a_1 \pmod{p_1} \\\\ + x &\equiv a_1 \pmod{m_1} \\\\ &\ldots \\\\ - x &\equiv a_k \pmod{p_k} + x &\equiv a_k \pmod{m_k} \end{align}$$ -(As above, assume that $p = p_1 p_2 \cdots p_k$ and $p_i$ are pairwise relatively prime). +(As above, assume that $m = m_1 m_2 \cdots m_k$ and $m_i$ are pairwise coprime). ## Garner's Algorithm pFad - Phonifier reborn

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