Question

VA ROGG, AM NEVOIE URGENT, VA ROGG, nu imi raspunde nimeni si sunt foarte ingrijorat, am nevoie va rog mult de tot. Ofer coroana​

Answer 1

Explicație pas cu pas:

a) a ≠ 0

$\overline {abc} \ \ \vdots \ \ 7 \implies (100a + 10b + c) \ \ \vdots \ \ 7$

$100a + 9b + b + c = 100a + 9b + a = 101a + 9b = \\ = 98a + 3a + 7b + 2b = 7 \cdot (14a + b) + (3a + 2b) \ \ \vdots \ \ 7$

$7 \cdot (14a + b) \ \ \vdots \ \ 7 \implies (3a + 2b) \ \ \vdots \ \ 7$

$c = a - b \iff a > b$

=>

$a = 4 \iff b = 1 \iff c = 3 \implies \overline {abc} = 413 \\ a = 5 \iff b = 3 \iff c = 2 \implies \overline {abc} = 532 \\ a = 6 \iff b = 5 \iff c = 1 \implies \overline {abc} = 651 \\ a = 7 \iff b = 0 \iff c = 7 \implies \overline {abc} = 707 \\ a = 7 \iff b = 7 \iff c = 0 \implies \overline {abc} = 770 \\ a = 8 \iff b = 2 \iff c = 6 \implies \overline {abc} = 826 \\ a = 9 \iff b = 4 \iff c = 5 \implies \overline {abc} = 945$

b) a ≠ 0

$\overline {abc} \ \ \vdots \ \ 7 \Leftrightarrow (a + b + c) \ \ \vdots \ \ 7$

$100a + 10b + c = 99a + 9b + (a + b + c) = 9(11 a + b) + (a + b + c)$

$\Big[9(11 a + b) + (a + b + c)\Big] \ \ \vdots \ \ 7 \implies 9 \cdot (11 a + b) \ \ \vdots \ \ 7$

$(9;7) = 1 \implies (11 a + b) \ \ \vdots \ \ 7$

=>

$a = 1 \iff b = 3 \iff c = 3 \implies \overline {abc} = 133 \\ a = 2 \iff b = 6 \iff c = 6 \implies \overline {abc} = 266 \\ a = 3 \iff b = 2 \iff c = 2 \implies \overline {abc} = 322 \\ a = 3 \iff b = 2 \iff c = 9 \implies \overline {abc} = 329 \\ a = 3 \iff b = 9 \iff c = 2 \implies \overline {abc} = 392 \\ a = 3 \iff b = 9 \iff c = 9 \implies \overline {abc} = 399 \\ a = 4 \iff b = 5 \iff c = 5 \implies \overline {abc} = 455 \\ a = 5 \iff b = 1 \iff c = 1 \implies \overline {abc} = 511 \\ a = 5 \iff b = 1 \iff c = 8 \implies \overline {abc} = 518 \\ a = 5 \iff b = 8 \iff c = 1 \implies \overline {abc} = 581 \\ a = 5 \iff b = 8 \iff c = 8 \implies \overline {abc} = 588 \\ a = 6 \iff b = 4 \iff c = 4 \implies \overline {abc} = 644 \\ a = 7 \iff b = 0 \iff c = 0 \implies \overline {abc} = 700 \\ a = 7 \iff b = 0 \iff c = 7 \implies \overline {abc} = 707 \\ a = 7 \iff b = 7 \iff c = 0 \implies \overline {abc} = 770 \\ a = 7 \iff b = 7 \iff c = 7 \implies \overline {abc} = 777 \\ a = 8 \iff b = 3 \iff c = 3 \implies \overline {abc} = 833 \\ a = 9 \iff b = 6 \iff c = 6 \implies \overline {abc} = 966$