99松山高中

99松山高中
例題  已知\(146 = {5^2} + {11^2}\)，\(218 = {7^2} + {13^2}\)，試將 \(146 \times 218 = 31828\)
表示成兩個正整數的平方和？
解:
\(146 \times 218 = {\left( t \right)^2} + {\left( k \right)^2}\)  \(t\)是正整數，\(k\)是正整數。

(1)  令 \(a = 5,b = 11,c = 7,d = 13\)
\[\begin{align}
  & 146\times 218=\left( {{a}^{2}}+{{b}^{2}} \right)\left( {{c}^{2}}+{{d}^{2}} \right) \\
& ={{a}^{2}}{{c}^{2}}+{{a}^{2}}{{d}^{2}}+{{b}^{2}}{{c}^{2}}+{{b}^{2}}{{d}^{2}} \\
& =\left\{ {{\left( ac \right)}^{2}}+2\left( ac \right)\left( bd \right)+{{\left( bd \right)}^{2}} \right\}+\left\{ {{\left( ad \right)}^{2}}+2\left( ad \right)\left( bc \right)+{{\left( bc \right)}^{2}} \right\} \\
& ={{\left\{ ac+bd \right\}}^{2}}+{{\left\{ ad-bc \right\}}^{2}} \\
& ={{\left( 5\times 7+11\times 13 \right)}^{2}}+{{\left( 5\times 13-11\times 7 \right)}^{2}} \\
& ={{178}^{2}}+{{\left( -12 \right)}^{2}} \\
\end{align}\]

答案  \(178\) 與 \(12\)

想法: 順著題目意思，把\(5,11,7,13\)改成\(a,b,c,d\)，比較好運算，接著配成完全平方。