Question

Determinaţi numărul n in fiecare dintre următoarele cazuri: ​

Answer 1

Explicație pas cu pas:

10.

$n = \overline {a1b} \ , \ \ 45 \ \Big| \ n \ , \ \ a \neq0$

$45 = 5 \cdot 9 \ , \ \ (5;9) = 1 \implies 5 \ \Big| \ n \ \ si \ \ 9 \ \Big| \ n$

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

$9 \ \Big| \ n \implies 9 \ \Big| \ (a + b + 1)$

$b = 0 \iff 9 \ \Big| \ (a + 1) \implies a = 8 \\ \implies \bf n = 810$

$b = 5 \iff 9 \ \Big| \ (a + 6) \implies a = 3 \\ \implies \bf n = 315$

11.

$n = \overline {2ab} \ , \ \ 15 \ \Big| \ n$

$15 = 5 \cdot 3 \ , \ \ (3;5) = 1 \implies 5 \ \Big| \ n \ \ si \ \ 3 \ \Big| \ n$

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

$3 \ \Big| \ n \implies 3 \ \Big| \ (a + b + 2)$

$b = 0 \iff 3 \ \Big| \ (a + 2) \implies a \in \Big\{ 1; 4; 7\Big\} \\ \implies \bf n \in \Big\{ 210; 240; 270\Big\}$

$b = 5 \iff 3 \ \Big| \ (a + 7) \implies a \in \Big\{ 2; 5; 8\Big\} \\ \implies \bf n \in \Big\{ 225; 255; 285\Big\}$

12.

$n = \overline {1a2b} \ , \ \ 18 \ \Big| \ n$

$18 = 2 \cdot 9 \ , \ \ (2;9) = 1 \implies 2 \ \Big| \ n \ \ si \ \ 9 \ \Big| \ n$

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

$9 \ \Big| \ n \implies 9 \ \Big| \ (a + b + 3)$

$\implies \bf n \in \Big\{1620; 1422; 1224; 1026; 1926; 1728 \Big\}$

13.

$n = \overline {3a5b} \ , \ \ 36 \ \Big| \ n$

$36 = 4 \cdot 9 \ , \ \ (4;9) = 1 \implies 4 \ \Big| \ n \ \ si \ \ 9 \ \Big| \ n$

$4 \ \Big| \ n \implies 4 \ \Big| \ \overline {5b} \implies b \in \Big\{ 2; 6\Big\}$

$9 \ \Big| \ n \implies 9 \ \Big| \ (a + b + 8)$

$b = 2 \iff 9 \ \Big| \ (a + 10) \implies a = 8 \\ \implies \bf n = 3852$

$b = 6 \iff 9 \ \Big| \ (a + 14) \implies a = 4 \\ \implies \bf n = 3456$

14.

$n = \overline {a7b} \ , \ \ 6 \ \Big| \ n \ , \ \ a \neq0$

$6 = 2 \cdot 3 \ , \ \ (2;3) = 1 \implies 2 \ \Big| \ n \ \ si \ \ 3 \ \Big| \ n$

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

$3 \ \Big| \ n \implies 3 \ \Big| \ (a + b + 7)$

$\implies \bf n \in \Big\{270; 570; 870; 372; 672; 972; 174; 474; 774; 276; 576; 876; 378; 678; 978\Big\}$

15.

$n = \overline {a32b} \ , \ \ 225 \ \Big| \ n$

$225 = 25 \cdot 9 \ , \ \ (25;9) = 1 \implies 25 \ \Big| \ n \ \ si \ \ 9 \ \Big| \ n$

$25 \ \Big| \ n \implies 25 \ \Big| \ \overline {2b} \implies b = 5$

$9 \ \Big| \ n \implies 9 \ \Big| \ (a + 10)\implies a = 8$

$\implies \bf n = 8325$