Question

Determină numărul abc stiind, că abc + bc + cb + 6•b = 379​

Answer 1

Răspuns:

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$\bf \boxed{\boxed{\bf \;\; \overline{abc}=193\;\; }}$

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Explicație pas cu pas:

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$\bf \overline{abc} +\overline{bc} +\overline{cb} + 6b=379$

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$\left.\begin{aligned} \bf \overline{abc} =100a+10b+c \\ \\ \bf \overline{bc}=10b+c \;\;\;\;\;\;\;\;\;\; \\ \\ \bf \overline{cb}=10c+b \;\;\;\;\;\;\;\;\;\; \end{aligned} \; \; \right\} \implies \bf \overline{abc} +\overline{bc} +\overline{cb} + 6b =$

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$\bf 100a+10b+c+10b+c+10c+b+6b=379$

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$\bf 100a + 27b+12c=379 \implies \overline{abc} <379$

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$\bf \implies a \in \{1;2;3\} \;\; \mathsf{Luam \; pe\; cazuri \; in \; functie \; de \; a:}$

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$\bf a=1 \; \Bigg| \implies 100a+27b+12c=100+27b+12c=379\; \Bigg|_{-100 }$

$\bf 27b+12c=279\; \Bigg| : 3 \iff 9b+4c=93 \; \;- convine$

$\bf \Big(27b\; ;12c\Big) \; \vdots \; \: 3 \implies \Big(9b+4c \Big) \; \vdots \;\: 3$

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$\left.\begin{aligned} \bf 4c\; \vdots \;\: 3 \implies 3 \in M_{4c} \\ \\\bf (4\; ; 3)=1-prime \end{aligned}\;\; \right\}\bf \implies c \: \; \vdots \;\: 3 \iff c \in M_{3}=\{3;6;9\}$

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$\bf c=3 \; \Bigg| \implies 9b+4c=9b+12=93 \Big| _{-12} \iff b=\dfrac{81}{9} =9$

$\bf \implies \boxed{\bf \overline{abc}=193}$

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$\bf c=6 \; \Bigg| \implies 9b+4c=9b+24=93\;\Big|_{-24} \iff b= \dfrac{69}{9} \notin \mathbb{N}$

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$\bf c=9\; \Bigg| \implies 9b+4c=9b+36=93\;\Big|_{-36} \iff b=\dfrac{57}{9} \notin \mathbb{N}$

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$\bf a=2 \; \Bigg| \implies 100a+27b+12c=200+27b+12c=379 \; \Bigg|_{-200}$

$\bf 27b+12c=179 \; \Bigg| : 3 \iff 9b+4c= \dfrac{179}{3} \; -nu \; convine$

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$\bf a=3 \; \Bigg| \implies 100a+27b+12c=300+27b+12c=379\; \Bigg|_{-300}$

$\bf 27b+12c=79\; \Bigg| : 3 \iff 9b+4c = \dfrac{79}{3} \; -nu\; convine$

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