S:P18.12: Difference between revisions
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Notăm prin <math>u\left( N \right)</math> ultima cifră a numărului natural <math>N</math>. | Notăm prin <math>u\left( N \right)</math> ultima cifră a numărului natural <math>N</math>. | ||
Avem | Avem <math>\overline{abc} \times 3 = \overline{bcc} \Rightarrow 300a+30b+3c=100b+11c \Rightarrow 300a = 70b + 8c</math>. | ||
Cum <math>u\left( 300 a\right) = 0</math> și <math>u\left( 70b \right) = 0</math>, se deduce că <math>u\left( 8c \right) = 0</math>, deci avem posibilitățile <math>c=0</math> și <math>c=5</math>. | |||
Pentru <math>c=0</math> se obține <math>b=0</math>, care nu convine. | |||
Pentru <math>c=5</math>, avem <math>\overline{abc} \times 3 = \overline{bcc} \Rightarrow 30a = 7b+4</math>. Cum <math>u\left( 30 a\right) = 0</math>, rezultă <math>u\left( 7b+4\right) = 0</math>, deci <math>u\left( 7b\right) = 6</math>, care implică <math>b=8</math>. Atunci avem <math>30a=56+4</math>, deci <math>a=2</math>. | |||
În concluzie, avem înmulțirea <math>285 \cdot 3 = 855</math>. | |||
Latest revision as of 12:18, 16 December 2024
S:P18.12 (Vasile Berinde)
Reconstituiți înmulțirea 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{abc} \times 3 = \overline{bcc}} .
Soluție.
Notăm prin 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 u\left( N \right)} ultima cifră a numărului 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 N} .
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{abc} \times 3 = \overline{bcc} \Rightarrow 300a+30b+3c=100b+11c \Rightarrow 300a = 70b + 8c} .
Cum 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 u\left( 70b \right) = 0} , se deduce că 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 u\left( 8c \right) = 0} , deci avem posibilitățile 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} 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=5} .
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 c=0} 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=0} , care nu convine.
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 c=5} , 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{abc} \times 3 = \overline{bcc} \Rightarrow 30a = 7b+4} . 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 u\left( 30 a\right) = 0} , rezultă 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 u\left( 7b+4\right) = 0} , deci 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 u\left( 7b\right) = 6} , care implică 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=8} . Atunci 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 30a=56+4} , deci 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} .
În concluzie, avem înmulțirea 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 285 \cdot 3 = 855} .