Question

Determinaţi toate numerele de forma abc, cu proprietatea: abc+9×(a+b+c) = cba​

Answer 1

Explicație pas cu pas:

a,b,c cifre în baza 10, a≠0, c≠0

$\overline {abc} +9 \cdot (a+b+c) = \overline {cba}$

$100a + 10b + c + 9a + 9b + 9c = 100c + 10b + a$

$109a - a = 100c - 10c + 10b - 19b \\ 108a = 90c + 9b \ \ | (: 9)| \\ \implies \bf 12a = 10c + b$

$1 \leqslant c \leqslant 9 \iff 10 \leqslant 10c \leqslant 90 \\ 0 \leqslant b \leqslant 9 \iff 10 \leqslant 10c + b \leqslant 99 \\ \implies 10 \leqslant 12a \leqslant 99 \implies \frac{10}{12} \leqslant a \leqslant \frac{99}{12} \\ \frac{5}{6} \leqslant a \leqslant 8\frac{1}{4} \iff 1 \leqslant a \leqslant 8$

$a = 1 \implies 12 = 10c + b \\ c = 1, b = 2 \implies \bf \overline {abc} = 121 \\ a = 2 \implies 24 = 10c + b \\ c = 2, b = 4 \implies \bf \overline {abc} = 242 \\ a = 3 \implies 36 = 10c + b \\ c = 3, b = 6 \implies \bf \overline {abc} = 363 \\ a = 4 \implies 48 = 10c + b \\ c = 4, b = 8 \implies \bf \overline {abc} = 484 \\ a = 5 \implies 60 = 10c + b \\ c = 6, b = 0 \implies \bf \overline {abc} = 506 \\ a = 6 \implies 72 = 10c + b \\ c = 7, b = 2 \implies \bf \overline {abc} = 627 \\ a = 7 \implies 84 = 10c + b \\ c = 8, b = 4 \implies \bf \overline {abc} = 748 \\ a = 8 \implies 96 = 10c + b \\ c = 9, b = 6 \implies \bf \overline {abc} = 869$