Răspuns: [tex]\boxed{\bf \overline{abc} \in \{ 197,298,399\}}[/tex]
Explicație pas cu pas:
Salut !
[tex]\bf \overline{abc} =?[/tex]
[tex]\text{\bf a,b,c - cifre }[/tex]
[tex]\text{\bf Cifrele sunt: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}[/tex]
[tex]\bf a\neq 0[/tex]
[tex]\bf \overline{ab}+\overline{bc}=\overline{ca}+45[/tex]
[tex]\text{\it Descompunem in baza 10 }[/tex]
[tex]\bf 10\cdot a+b+10\cdot b+c=10\cdot c+a+45[/tex]
[tex]\bf 10a+11b+c = 10c +a + 45[/tex]
[tex]\bf 10a+11b+c-10c-a = 45[/tex]
[tex]\bf 9a+11b-9c=45[/tex]
[tex]\bf 9\cdot(a-c)+11b=45[/tex]
↓ ↓
[tex]\bf \vdots\:\: 9[/tex] [tex]\bf \vdots\:\: 9[/tex]
[tex]\text{\it 9(a-c) si 45 sunt divizibile cu 9}\implies \text{\it 11b e divizibil cu 9}\implies (9;11)=1[/tex]
[tex]\text{\bf b - cifra}\implies \text{\bf b = 9}[/tex]
[tex]\bf 9(a-c) + 11\cdot 9 = 45\:\:\:\:\Big|:9\:\:\text{\it(Impartim toata relatia cu 9)}[/tex]
[tex]\bf (a-c) + 11 = 5[/tex]
[tex]\bf a-c + 11 = 5[/tex]
[tex]\bf a + 11 = 5 +c[/tex]
[tex]\bf a + 11-5=c[/tex]
[tex]\bf a +6 =c[/tex]
[tex]\text{\it Vom observa ca c poate avea maxim valoarea 9, astfel vom da valori lui a}[/tex]
- [tex]\bf\underline{a=1} \Rightarrow 1+6 = c \Rightarrow \underline{c = 7} \implies \underline{abc = 197}[/tex]
- [tex]\bf\underline{a=2} \Rightarrow 2+6 = c \Rightarrow \underline{c = 8} \implies \underline{abc = 298}[/tex]
-
[tex]\bf\underline{a=3} \Rightarrow 3+6 = c \Rightarrow \underline{c = 9} \implies \underline{abc = 399}[/tex]
[tex]\boxed{\bf \overline{abc} \in \{ 197,298,399\}}[/tex]
[tex]\it Verificare:[/tex]
[tex]\it 19 + 97 = 71 + 45 \:(adevarat)[/tex]
[tex]\it 29 + 98 = 82 + 45 \:(adevarat)[/tex]
[tex]\it 39 + 99 = 93 + 45 \:(adevarat)[/tex]
==pav38==
