Question

aflati cel mai mic si cel mai mare numar de 3 cifre diferite scris in baza 10 care prin eliminarea cifrei din ijloc devine de 11 ori mai mic decat numarul initial

Answer 1

   
[tex]\overline{abc} = 11 \times \overline{ac} \\ 100a + 10b + c = 11(10a + c)\\ 100a + 10b + c = 110a + 11c\\ 10b = (110a - 100a) + (11c - c)\\ 10b = 10a + 10c\\ 10b = 10(a+c) | : 10\\ b = a+c\\ \\ [/tex]


[tex]\displaystyle \\ \text{Cautam numarul cel mai mic.} \\ \\ \text{Cea mai mica valoare pentru b:} \\ b = 1;~~~ a+c = 1;~~ \Longrigharrow~~ \overline{abc} = 110;~~a=b;~~\text{Respins} \\ \\ b = 2;~~~ a+c = 2;~~ \Longrigharrow~~ \overline{abc} = 121;~~a=c;~~\text{Respins} \\ \\ b = 3;~~~ a+c = 3;~~ \Longrigharrow~~ \overline{abc} = \boxed{132}\\ \\ \texttt{Verificare: }~~ \boxed{\frac{132}{12} = 11} [/tex]


[tex]\displaystyle \\ \text{Cautam numarul cel mai mare.} \\ \\ \text{Cea mai mare valoare pentru b:} \\ b = 9;~~~ a+c = 9;~~ \Longrigharrow~~ \overline{abc} = 990;~~a=b;~~\text{Respins} \\ \\ b = 9;~~~ a+c = 9;~~ \Longrigharrow~~ \overline{abc} = \boxed{891}\\ \\ \texttt{Verificare: }~~ \boxed{\frac{891}{81} = 11} [/tex]