P:1792

P:1792 (Monica Dragoș)

Determinați numărul natural 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{ab}} pentru care 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 \times \overline{b00b} + \overline{aa} = 2024} .

Soluție

Evident, cifrele $a$ și $b$ sunt diferite de $0$.

Din ${\overline {b00b}}<2024$ se obține $b\leq 2$, deci cifra $b$ poate lua valorile $1$ sau $2$.

Egalitatea din enunț se poate scrie în forma echivalentă $a\times b\times 1001+a\times 11=2024$. Cum $1001=11\times 91$ și $2024=11\times 184$, fiecare termen se împarte cu $11$ și se obține $a\times b\times 91+a=184$.

Pentru $b=1$, egalitatea de mai sus 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 91 \times a + a = 184} , ceea ce conduce la 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 92 \times a = 184} , ecuație ce are soluția 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} .

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 b=2} , egalitatea de mai sus 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 182 \times a + a = 184} , ceea ce conduce la egalitatea 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 183 \times a = 184} pentru care nu există 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} număr natural care să o satisfacă.