P:1792: Difference between revisions
Created page with "'''P:1792 (Monica Dragoș)''' ''Determinați numărul natural <math>\overline{ab}</math> pentru care <math>a \times \overline{b00b} + \overline{aa} = 2024</math>.'' '''Soluție''' Evident, cifrele <math>a</math> și <math>b</math> sunt diferite de <math>0</math>. Din <math>\overline{b00b} < 2024</math> se obține <math>b\le2</math>, deci cifra <math>b</math> poate lua valorile <math>1</math> sau <math>2</math>. Egalitatea din enunț se poate scrie în forma echivale..." |
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Pentru <math>b=1</math>, egalitatea de mai sus devine <math>91 \times a + a = 184</math>, ceea ce conduce la <math>92 \times a = 184</math>, ecuație ce are soluția <math>a=2</math>. | Pentru <math>b=1</math>, egalitatea de mai sus devine <math>91 \times a + a = 184</math>, ceea ce conduce la <math>92 \times a = 184</math>, ecuație ce are soluția <math>a=2</math>. | ||
Pentru <math>b=2</math>, egalitatea de mai sus devine <math>182 \times a + a = 184</math>, ceea ce conduce la egalitatea <math>183 \times a = 184</math> pentru care nu există <math>a</math> număr natural care să o satisfacă. | Pentru <math>b=2</math>, egalitatea de mai sus devine <math>182 \times a + a = 184</math>, ceea ce conduce la egalitatea <math>183 \times a = 184</math> pentru care nu există <math>a</math>, număr natural, care să o satisfacă. | ||
În concluzie, avem <math>\overline{ab}=21</math>. | |||
Latest revision as of 12:23, 16 August 2025
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 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} 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 b} sunt diferite de .
Din se obține 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\le2} , deci cifra 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} poate lua valorile 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} 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 2} .
Egalitatea din enunț se poate scrie în forma echivalentă 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 b \times 1001 + a \times 11 =2024} . 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 1001 = 11 \times 91} 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 2024 = 11 \times 184} , fiecare termen se împarte cu 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 11} și se obține 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 b \times 91 + a = 184} .
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=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.
În concluzie, 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 \overline{ab}=21} .