E:15694

E:15694 (Traian Covaciu, Baia Mare)

Suma a două numere naturale nenule este 2020. Dacă împărţim primul număr la al doilea, obţinem câtul egal cu restul. Aflaţi cele două numere.

Soluție.

Dacă $a$ şi $b$ sunt cele două numere, $a>b$ şi $c$ este câtul şi restul, atunci $a+b=2020$ $(1)$ şi $a=b\cdot c+c$ $(2)$, cu $c<b$. Înlocuind pe $(2)$ în $(1)$ avem $b\cdot c+c+b=2020$. sau $c(b+1)+b+1=2021$, de unde $(b+1)(c+1)=2021$. Cum $2021=43\cdot 47$ putem avea 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 + 1 = 43} şi 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 + 1 = 47} 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 b + 1 = 47} şi 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 + 1 = 43} . În primul caz 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 = 42} şi 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 = 46} , care nu convine (avem condiţia 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 < b} ). În al doilea caz 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 = 46} şi 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 = 42} . Obţinem 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 = 1974} .