E:16382

E:16382 (Cristina Vijdeluc și Mihai Vijdeluc)

Afișați numerele întregi pozitive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abcd}} cu proprietatea

$$a^{7}+a^{b}+a^{c}+a^{d}={\overline {a000}}.$$

Soluție

Deoarece $4^{7}>9999$ și $1^{7}+1^{b}+1^{c}+1^{d}<1000$, avem $a=2$ sau $a=3$.

Egalitatea din enunț corespunde scrierii numărului ${\overline {a000}}$ în baza Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} .

Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2000 = 11111010000_2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=2} nu convine.

De asemenea, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3000 = 101010010_3} corespunde egalității din enunț Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3000 = 3^7 + 3^5 + 3^4 + 3^2 \cdot 2^2.} Prin urmare, avem soluțiile Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 6} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = 4} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 1} și permutările acestora.

Se obțin numerele întregi pozitive: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{{3641}, {3614}, {3461}, {3416}, {3164}, {3146}\}.}