15678

15678 (Cristina Vijdeluc și Mihai Vijdeluc)

Aflați toate numerele de forma 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}} pentru careFailed 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} = 2021+5(a-c+b-d+1).}

Soluție: Ultima cifră 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 5(a-c+b-d+1)} poate fi $0$ sau $5$. Atunci ultima cifră a numărului $2021+5(a-c+b-d+1)$ poate fi $1$ sau $6$, de unde deducem că $d=1$ sau $d=6$.

1. Pentru $d=1$ relația devine ${\overline {abcd}}=2021+5(a-c+b)$. Deoarece $a-c+b\leq 18$ avem $2021+5(a-c+b)\leq 2111$ și, cum ${\overline {abcd}}=2021+5(a-c+b)$, obținem $a\leq 2$.

Pentru $a=2$ relația dată devine ${\overline {2bc1}}=2021+5(2-c+b)$ sau $2001+100b+10c=2031-5c+5b$, de unde $95b+15c=30$.

De aici $b=0$ și $c=2$. Numărul căutat este $2021$.

Pentru $a=1$ nu obținem nicio soluție.

2. Pentru $d=6$ relația dată devine ${\overline {abc6}}=2021+5(a-c+b-5)$. Deoarece $a-c+b-5\leq 13$ avem $2021+5(a-c+b)\leq 2086$ și 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 \overline{abc6} = 2021+5(a-c+b-5)} , obținem 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 \le 2} .

Pentru 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} relația dată devine 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{2bc6} = 2021+5(2-c+b-5)} 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 2006+100b+10c=2006-5c+5b} , de unde 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 95b+15c=0} . De aici 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=0} 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 c=0} . Numărul căutat este 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 2006} .

Pentru 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=1} nu obținem nicio soluție.