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Question Number 189302 by mnjuly1970 last updated on 14/Mar/23

$$$$$$Q:\mathrm{find}\mathrm{the}\mathrm{number}\mathrm{of}$$$$\mathrm{the}\mathrm{solutions}\mathrm{for}:$$$$$$$${(x_1+x_2)}^3+x_3+x_4+x_5=11$$$$$$$$Hint:(x_i\in {\mathbb{Z}}^{+}\cup \left\{0\right\})$$$$$$

Commented by Frix last updated on 14/Mar/23

$${\mathbb{Z}}^{+}\cup \left\{0\right\}=\mathbb{N}$$

Commented by Frix last updated on 14/Mar/23

$$a^3+b=11$$$$a=0\wedge b=11$$$$a=1\wedge b=10$$$$a=2\wedge b=3$$$$\mathrm{Now}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{just}\mathrm{counting}...$$

Answered by manxsol last updated on 15/Mar/23

$$a^3+b=11$$$$\overline{)\begin{array}{ccc}a& b& \\ 0& 11& \\ 1& 10& \\ 2& 3& \end{array}}{x}_{c_{}}=\underset{4}{x}+x_5$$$$\overline{)\begin{array}{ccc}a& x_1& x_2\\ 0& 0& 0\\ 1& 0& 1\\ 1& 1& 0\\ 2& 0& 2\\ 2& 1& 1\\ 2& 2& 0\end{array}}\overline{)\begin{array}{cccc}b& x_3& x_c& \\ 11& 11& 0& \\ 11& 10& 1& \\ 11& 9& 2& \\ & & & \\ & & & \\ 11& 0& 11& \end{array}}$$$$$$$${\overline{)\begin{array}{cccc}{x}_c& x_4& x_5& \mathrm{♯}\\ 0& 0& 0& 1\\ 1& \overline{)\begin{array}{c}0\\ 1\end{array}}& \overline{)\begin{array}{c}1\\ 0\end{array}}& 2\\ 2& \overline{)\begin{array}{c}0\\ 1\\ 2\end{array}}& \overline{)\begin{array}{c}2\\ 1\\ 2\end{array}}& 3\\ 3& \overline{)\begin{array}{c}0\\ 1\\ 2\\ 3\end{array}}& \overline{)\begin{array}{c}3\\ 2\\ 1\\ 0\end{array}}& 4\\ & & & \\ 4& & & \\ 5& & & \\ 6& & & \\ 7& & & \\ 8& & & \\ 9& & & \\ 10& & & \\ 11& & & \\ & & & \sum _1^{12}=78\end{array}}}_{}$$$$\overline{)\begin{array}{cccc}{x}_1{x}_2& x_3& x_4{x}_5& (x_{\hat{1}}{x}_2,x_{3,},x_4,x_5\\ 6& 12& 78& 5616\end{array}}$$

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