回復 2# simon112266 的帖子
第 2 題：
\(36 = a\cdot 7\cdot 11 + b\cdot 5\cdot 11+c\cdot5\cdot 7\)
可知
\(\displaystyle \left\{\begin{array}{cc}36\equiv a\cdot 7\cdot 11\pmod{5}\\ 36\equiv b\cdot 5\cdot 11\pmod{7} \\ 36\equiv c\cdot5\cdot 7\pmod{11}\end{array}\right.\)
\(\displaystyle \Rightarrow\left\{\begin{array}{cc}1\equiv 2a\pmod{5}\\ 1\equiv 6b\pmod{7} \\ 3\equiv 2c\pmod{11}\end{array}\right.\)
\(\displaystyle \Rightarrow\left\{\begin{array}{cc}3\cdot1\equiv 3\cdot2a\pmod{5}\\ 6\cdot1\equiv 6\cdot6b\pmod{7} \\ 6\cdot3\equiv 6\cdot2c\pmod{11}\end{array}\right.\)
\(\displaystyle \Rightarrow\left\{\begin{array}{cc}3\equiv a\pmod{5}\\ 6\equiv b\pmod{7} \\ 7\equiv c\pmod{11}\end{array}\right.\)
且由 \(\left|a\right|<5, \left|b\right|<7, \left|c\right|<11\)
可知 \(a=3\) 或 \(a=-2\)，\(b=6\) 或 \(b=-1\)，\(c=7\) 或 \(c=-4\)，
剩下就是帶入 \(\displaystyle \frac{36}{385}=\frac{a}{5}+\frac{b}{7}+\frac{c}{11}\)
檢查可得正確的有序數組 \(\left(a,b,c\right)=(3,-1,-4), (-2,6,-4), (-2,-1,7)\)
