回復 1# satsuki931000 的帖子
第9題
正整數\(n\)是合數，將\(n\)的正因數由小而大依序記為\(d_1,d_2,d_3,\ldots,d_n\)，設\(f(n)=d_1+d_2+d_3,g(n)=d_{n-1}+d_n\)，若\(g(n)=(f(n))^3\)，試求所有可能的正整數\(n\)。
這題題目出得不好
應是將\(n\)的正因數由小而大依序記為\({{d}_{1}},{{d}_{2}},{{d}_{3}},\cdots ,{{d}_{k}}\)
然後\(g\left( n \right)={{d}_{k-1}}+{{d}_{k}}\)
\({{d}_{1}}=1,{{d}_{k}}=n\)
(1)\({{d}_{2}}=2\)
\(\begin{align}
& \frac{n}{2}+n=g\left( n \right)={{\left( f\left( n \right) \right)}^{3}}={{\left( 1+2+{{d}_{3}} \right)}^{3}} \\
& n={{\left( 3+{{d}_{3}} \right)}^{3}}\times \frac{2}{3} \\
& {{d}_{3}}=3,n=144 \\
\end{align}\)
其餘不合
(2)\({{d}_{2}},{{d}_{3}},\cdots \cdots ,{{d}_{k}}\)均為奇數
\(\begin{align}
& \frac{n}{{{d}_{2}}}+n=g\left( n \right)={{\left( f\left( n \right) \right)}^{3}}={{\left( 1+{{d}_{2}}+{{d}_{3}} \right)}^{3}} \\
& n={{\left( 1+{{d}_{2}}+{{d}_{3}} \right)}^{3}}\times \frac{{{d}_{2}}}{1+{{d}_{2}}} \\
\end{align}\)
\({{\left( 1+{{d}_{2}}+{{d}_{3}} \right)}^{3}}\)為奇數，\(1+{{d}_{2}}\)為偶數，不合
