Question

5. Determinaţi numărul n în fiecare dintre următoarele cazuri: a) n = 2ab, 5|n şi 3|n;
b) n=a32b, 2|n şi 9|n;
c) n=alb, 10|n şi 3|n;
d) n = 2abc, 10²|n şi 9 n.

(coroana)​

Answer 1

Explicație pas cu pas:

a)

$n = \overline {2ab} \ , \ 5 \mid n \ , \ 3 \mid n$

$5 \mid n \implies b \in \Big\{ 0; 5\Big\}$

$3 \mid n \iff 3 \mid (2 + a + b)$

$b = 0 \iff 3 \mid (2 + a) \implies a \in \Big\{ 1; 4; 7 \Big\} \\ b = 5 \iff 3 \mid (2 + 5) \implies a \in \Big\{ 2; 5; 8 \Big\}$

$n \in \Big\{ 210; 240; 270; 225; 255; 285\Big\}$

b)

$n = \overline {a32b} \ , \ 2 \mid n \ , \ 9 \mid n$

$2 \mid n \implies b \in \Big\{ 0; 2; 4; 6; 8 \Big\}$

$9 \mid n \iff 9 \mid (a + 3 + 2 + b) \iff 9 \mid (a + b + 5)$

$b = 0 \iff 9 \mid (a + 5) \ \ \implies a = 4 \\ b = 2 \iff 9 \mid (a + 7) \ \ \implies a = 2 \\ b = 4 \iff 9 \mid (a + 9) \ \ \implies a = 9 \\ b = 6 \iff 9 \mid (a + 11) \implies a = 7 \\ b = 8 \iff 9 \mid (a + 13) \implies a = 5$

$n \in \Big\{4320; 2322; 9324; 7326; 5328\Big\}$

c)

$n = \overline {a1b} \ , \ 10 \mid n \ , \ 3 \mid n$

$10 \mid n \implies b = 0$

$3 \mid n \iff 3 \mid (a + 1) \implies a \in \Big\{2; 5; 8 \Big\}$

$n \in \Big\{210; 510; 810\Big\}$

d)

$n = \overline {2abc} \ , \ {10}^{2} \mid n \ , \ 9 \mid n$

${10}^{2} \mid n \implies b = 0 \ , \ c = 0$

$9 \mid n \implies 9 \mid (2 + a + b + c) \\ \iff 9 \mid (2 + a) \implies \ a = 7$

$n = 2700$