Question

Determinati nr de forma abc care verifica relatia ab+bc=ca+45​

Answer 1

Răspuns: $\boxed{\bf \overline{abc} \in \{ 197,298,399\}}$

Explicație pas cu pas:

Salut !

$\bf \overline{abc} =?$

$\text{\bf a,b,c - cifre }$

$\text{\bf Cifrele sunt: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$

$\bf a\neq 0$

$\bf \overline{ab}+\overline{bc}=\overline{ca}+45$

$\text{\it Descompunem in baza 10 }$

$\bf 10\cdot a+b+10\cdot b+c=10\cdot c+a+45$

$\bf 10a+11b+c = 10c +a + 45$

$\bf 10a+11b+c-10c-a = 45$

$\bf 9a+11b-9c=45$

$\bf 9\cdot(a-c)+11b=45$

    ↓                          ↓

   $\bf \vdots\:\: 9$                        $\bf \vdots\:\: 9$  

$\text{\it 9(a-c) si 45 sunt divizibile cu 9}\implies \text{\it 11b e divizibil cu 9}\implies (9;11)=1$

$\text{\bf b - cifra}\implies \text{\bf b = 9}$

$\bf 9(a-c) + 11\cdot 9 = 45\:\:\:\:\Big|:9\:\:\text{\it(Impartim toata relatia cu 9)}$

$\bf (a-c) + 11 = 5$

$\bf a-c + 11 = 5$

$\bf a + 11 = 5 +c$

$\bf a + 11-5=c$

$\bf a +6 =c$

$\text{\it Vom observa ca c poate avea maxim valoarea 9, astfel vom da valori lui a}$

• $\bf\underline{a=1} \Rightarrow 1+6 = c \Rightarrow \underline{c = 7} \implies \underline{abc = 197}$

• $\bf\underline{a=2} \Rightarrow 2+6 = c \Rightarrow \underline{c = 8} \implies \underline{abc = 298}$

• $\bf\underline{a=3} \Rightarrow 3+6 = c \Rightarrow \underline{c = 9} \implies \underline{abc = 399}$

$\boxed{\bf \overline{abc} \in \{ 197,298,399\}}$

$\it Verificare:$

$\it 19 + 97 = 71 + 45 \:(adevarat)$

$\it 29 + 98 = 82 + 45 \:(adevarat)$

$\it 39 + 99 = 93 + 45 \:(adevarat)$                

==pav38==