E:16382

E:16382 (Cristina Vijdeluc și Mihai Vijdeluc)

Afișați numerele întregi pozitive ${\overline {abcd}}$ cu proprietateaFailed 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^7 + e^d + ad = \overline{a000}.} Soluție Deoarece 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 4^7 > 9999} și 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 1^7 + a^b + a^c + a^d < 1000} , avem 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} sau 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 = 3} . Egalitatea din enunț corespunde scrierii numărului 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{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 $2000=11111010000_{2}$, $a=2$ nu convine.

De asemenea, $3000=10101001000_{2}$ corespunde egalității din enunț

$$3000=3^{7}+3^{5}+3^{4}+3^{2}\cdot 2^{2}.$$

Prin urmare, avem soluțiile $b=6$, $c=4$, $d=1$ și permutările acestora.

Se obțin numerele întregi pozitive:

$$\{{3641},{3614},{3461},{3461},{3164},{3146}\}.$$