Răspuns: [tex]\bf \overline{10x0x} =10404[/tex]
Explicație pas cu pas:
Salutare !
[tex]\bf \overline{10x0x} =?[/tex]
[tex]\bf x - cifra[/tex]
[tex]\bf x\in \{0,1,2,3,4,5,6,7,8,9\}[/tex]
Criteriul de divizibilitate cu 9: "Un număr este divizibil cu 9 daca si numai daca suma cifrelor numărului respectiv se divide cu 9"
[tex]\bf \implies (1+0+x+0+x) \in M_{9} \implies (1+2x)\in \{9, 18 \}[/tex]
[tex]\bf 1)~Daca~ \underline{x = 1} \implies 1+2 \cdot 1 = 3 ~nu~ convine[/tex]
[tex]\bf 2)~Daca~ \underline{x = 2} \implies 1+2 \cdot2 = 1+4=5~\not\vdots~9[/tex]
[tex]\bf 3)~Daca~ \underline{x = 3} \implies 1+2 \cdot3 = 1+6=7~ \not\vdots~9[/tex]
[tex]\bf 4)~Daca~ \underline{x = 4} \implies 1+2 \cdot4 = 1+8=9~\vdots~9 \implies 10404~solutie[/tex]
[tex]\bf 5)~Daca~ \underline{x = 5} \implies 1+2 \cdot5 = 1+10=11~\not\vdots~9[/tex]
[tex]\bf 6)~Daca~\underline{x=6} \implies 1+2 \cdot6 = 1+12=13~\not\vdots~9[/tex]
[tex]\bf 7)~Daca~\underline{x=7} \implies 1+2 \cdot7 = 1+14=15~\not\vdots~9[/tex]
[tex]\bf 8)~Daca~\underline{x=8} \implies 1+2 \cdot8 = 1+16=17~\not\vdots~9[/tex]
[tex]\bf 9)~Daca~\underline{x=9} \implies 1+2 \cdot9 = 1+18=19~\not\vdots~9[/tex]
Din cazurile analizate x = 4 ⇒ [tex]\bf \overline{10x0x} =10404[/tex]
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