14以下的質數有3,5,7,11,13,故x的一個可能為
\(\left\{ \begin{array}{l}
x \equiv 3(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 10)\\
x \equiv 5(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 12)\\
x \equiv 7(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 14)
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x \equiv 3(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 5)\\
x \equiv 2(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 3)\\
x \equiv 0(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 7)\\
x \equiv 1(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 4)
\end{array} \right. \Leftrightarrow x \equiv ( - 84) \cdot 3 + ( - 140) \cdot 2 + 120 \cdot 0 + 105 \cdot 1 \equiv 413(\bmod {\kern 1pt} {\kern 1pt} 420)\)
更一般的情況可寫為
\(\left\{ \begin{array}{l}
x \equiv a(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 5)\\
x \equiv b(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 3)\\
x \equiv c(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 7)\\
x \equiv d(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 4)
\end{array} \right. \Leftrightarrow x \equiv ( - 84) \cdot a + ( - 140) \cdot b + 120 \cdot c + 105 \cdot d(\bmod {\kern 1pt} {\kern 1pt} 420)\)
模10 | 模12 | 模14 | 模5同餘a | 模3同餘b | 模7同c | 模4同餘d | x=-84a-140b+120c+105d 模420 |
3 | 5 | 7 | 3 | 2 | 0 | 1 | 413 |
3 | 5 | 11 | 3 | 2 | 4 | 1 | 53 |
3 | 5 | 13 | 3 | 2 | 6 | 1 | 293 |
3 | 7 | 5 | 3 | 1 | 5 | 3 | 103 |
3 | 7 | 11 | 3 | 1 | 4 | 3 | 403 |
3 | 7 | 13 | 3 | 1 | 6 | 3 | 223 |
3 | 11 | 5 | 3 | 2 | 5 | 3 | 383 |
3 | 11 | 7 | 3 | 2 | 0 | 3 | 203 |
3 | 11 | 13 | 3 | 2 | 6 | 3 | 83 |
5 | 3 | 7 | 0 | 0 | 0 | 3 | 315 |
5 | 3 | 11 | 0 | 0 | 4 | 3 | 375 |
5 | 3 | 13 | 0 | 0 | 6 | 3 | 195 |
5 | 7 | 3 | 0 | 1 | 3 | 3 | 115 |
5 | 7 | 11 | 0 | 1 | 4 | 3 | 235 |
5 | 7 | 13 | 0 | 1 | 6 | 3 | 55 |
5 | 11 | 3 | 0 | 2 | 3 | 3 | 395 |
5 | 11 | 7 | 0 | 2 | 0 | 3 | 35 |
5 | 11 | 13 | 0 | 2 | 6 | 3 | 335 |
7 | 3 | 5 | 2 | 0 | 5 | 3 | 327 |
7 | 3 | 11 | 2 | 0 | 4 | 3 | 207 |
7 | 3 | 13 | 2 | 0 | 6 | 3 | 27 |
7 | 5 | 3 | 2 | 2 | 3 | 1 | 17 |
7 | 5 | 11 | 2 | 2 | 4 | 1 | 137 |
7 | 5 | 13 | 2 | 2 | 6 | 1 | 377 |
7 | 11 | 3 | 2 | 2 | 3 | 3 | 227 |
7 | 11 | 5 | 2 | 2 | 5 | 3 | 47 |
7 | 11 | 13 | 2 | 2 | 6 | 3 | 167 |