Question Number 189302 by mnjuly1970 last updated on 14/Mar/23
$$ \\ $$$$\:\:\:\:{Q}:\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{the}\:\:\mathrm{solutions}\:\:\mathrm{for}\:: \\ $$$$ \\ $$$$\:\:\left(\:{x}_{\:\mathrm{1}} \:+\:{x}_{\:\mathrm{2}} \:\right)^{\:\mathrm{3}} \:+\:{x}_{\:\mathrm{3}} \:+\:{x}_{\:\mathrm{4}} \:+\:{x}_{\:\mathrm{5}} \:=\mathrm{11} \\ $$$$\:\:\: \\ $$$$\:\:\:\:{Hint}:\:\:\:\left(\:{x}_{\:{i}} \:\:\in\:\:\mathbb{Z}^{\:\:+} \:\:\cup\:\left\{\:\mathrm{0}\:\right\}\:\:\right) \\ $$$$\: \\ $$
Commented by Frix last updated on 14/Mar/23
$$\mathbb{Z}^{+} \cup\left\{\mathrm{0}\right\}=\mathbb{N} \\ $$
Commented by Frix last updated on 14/Mar/23
$${a}^{\mathrm{3}} +{b}=\mathrm{11} \\ $$$${a}=\mathrm{0}\wedge{b}=\mathrm{11} \\ $$$${a}=\mathrm{1}\wedge{b}=\mathrm{10} \\ $$$${a}=\mathrm{2}\wedge{b}=\mathrm{3} \\ $$$$\mathrm{Now}\:\mathrm{it}'\mathrm{s}\:\mathrm{just}\:\mathrm{counting}… \\ $$
Answered by manxsol last updated on 15/Mar/23
$${a}^{\mathrm{3}} +{b}=\mathrm{11} \\ $$$$\begin{array}{|c|c|c|c|}{{a}}&\hline{{b}}&\hline{}\\{\mathrm{0}}&\hline{\mathrm{11}}&\hline{}\\{\mathrm{1}}&\hline{\mathrm{10}}&\hline{}\\{\mathrm{2}}&\hline{\mathrm{3}}&\hline{}\\\hline\end{array}\:\:\:\:\:\:{x}_{{c}_{} } =\underset{\mathrm{4}} {{x}}+{x}_{\mathrm{5}} \\ $$$$\begin{array}{|c|c|c|c|c|c|c|}{{a}}&\hline{{x}_{\mathrm{1}} }&\hline{{x}_{\mathrm{2}} }\\{\mathrm{0}}&\hline{\mathrm{0}}&\hline{\mathrm{0}}\\{\mathrm{1}}&\hline{\mathrm{0}}&\hline{\mathrm{1}}\\{\mathrm{1}}&\hline{\mathrm{1}}&\hline{\mathrm{0}}\\{\mathrm{2}}&\hline{\mathrm{0}}&\hline{\mathrm{2}}\\{\mathrm{2}}&\hline{\mathrm{1}}&\hline{\mathrm{1}}\\{\mathrm{2}}&\hline{\mathrm{2}}&\hline{\mathrm{0}}\\\hline\end{array}\:\begin{array}{|c|c|c|c|c|c|c|}{{b}}&\hline{{x}_{\mathrm{3}} }&\hline{{x}_{{c}} }&\hline{}\\{\mathrm{11}}&\hline{\mathrm{11}}&\hline{\mathrm{0}}&\hline{}\\{\mathrm{11}}&\hline{\mathrm{10}}&\hline{\mathrm{1}}&\hline{}\\{\mathrm{11}}&\hline{\mathrm{9}}&\hline{\mathrm{2}}&\hline{}\\{}&\hline{}&\hline{}&\hline{}\\{}&\hline{}&\hline{}&\hline{}\\{\mathrm{11}}&\hline{\mathrm{0}}&\hline{\mathrm{11}}&\hline{}\\\hline\end{array} \\ $$$$ \\ $$$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}{{x}_{{c}} }&\hline{{x}_{\mathrm{4}} }&\hline{{x}_{\mathrm{5}} }&\hline{\sharp}\\{\mathrm{0}}&\hline{\mathrm{0}}&\hline{\mathrm{0}}&\hline{\mathrm{1}}\\{\mathrm{1}}&\hline{\begin{array}{|c|c|}{\mathrm{0}}\\{\mathrm{1}}\\\hline\end{array}}&\hline{\begin{array}{|c|c|}{\mathrm{1}}\\{\mathrm{0}}\\\hline\end{array}}&\hline{\mathrm{2}}\\{\mathrm{2}}&\hline{\begin{array}{|c|c|c|}{\mathrm{0}}\\{\mathrm{1}}\\{\mathrm{2}}\\\hline\end{array}}&\hline{\begin{array}{|c|c|c|}{\mathrm{2}}\\{\mathrm{1}}\\{\mathrm{2}}\\\hline\end{array}}&\hline{\mathrm{3}}\\{\mathrm{3}}&\hline{\begin{array}{|c|c|c|c|}{\mathrm{0}}\\{\mathrm{1}}\\{\mathrm{2}}\\{\mathrm{3}}\\\hline\end{array}}&\hline{\begin{array}{|c|c|c|c|}{\mathrm{3}}\\{\mathrm{2}}\\{\mathrm{1}}\\{\mathrm{0}}\\\hline\end{array}}&\hline{\mathrm{4}}\\{}&\hline{}&\hline{}&\hline{}\\{\mathrm{4}}&\hline{}&\hline{}&\hline{}\\{\mathrm{5}}&\hline{}&\hline{}&\hline{}\\{\mathrm{6}}&\hline{}&\hline{}&\hline{}\\{\mathrm{7}}&\hline{}&\hline{}&\hline{}\\{\mathrm{8}}&\hline{}&\hline{}&\hline{}\\{\mathrm{9}}&\hline{}&\hline{}&\hline{}\\{\mathrm{10}}&\hline{}&\hline{}&\hline{}\\{\mathrm{11}}&\hline{}&\hline{}&\hline{}\\{}&\hline{}&\hline{}&\hline{\sum_{\mathrm{1}} ^{\mathrm{12}} =\mathrm{78}}\\\hline\end{array}_{} \\ $$$$\begin{array}{|c|c|}{{x}_{\mathrm{1}\:\:} {x}_{\mathrm{2}} }&\hline{{x}_{\mathrm{3}} }&\hline{{x}_{\mathrm{4}\:\:\:} {x}_{\mathrm{5}} }&\hline{\left({x}_{\hat {\mathrm{1}}} {x}_{\mathrm{2}} ,{x}_{\mathrm{3},} ,{x}_{\mathrm{4}} ,{x}_{\mathrm{5}} \right.}\\{\mathrm{6}}&\hline{\mathrm{12}}&\hline{\mathrm{78}}&\hline{\mathrm{5616}}\\\hline\end{array} \\ $$