Question

Aflati toate nr naturale de trei cifre imaprtit la 35 ,42 si 70 dau de fiecare data restul 3​

Answer 1

Explicație pas cu pas:

$\begin{cases}n = c_{1} \cdot 35 + 3 \iff n - 3 = c_{1} \cdot 35 \\ n = c_{2} \cdot 42 + 3 \iff n - 3 = c_{2} \cdot 42 \\ n = c_{3} \cdot 70 + 3 \iff n - 3 = c_{3} \cdot 70\end{cases}$

calculăm c.m.m.m.c.:

35 = 5×7

42 = 2×3×7

70 = 2×5×7

[35;42;70] = 2×3×5×7 = 210

$(n - 3) \in \mathcal{M}_{210}$

$100 \leqslant n \leqslant 999 \iff 97 \leqslant n - 3 \leqslant 996$

$(n - 3) \in \Big\{ 210; 420; 630; 840\Big\} \\ \implies n \in \Big\{ 207; 417; 627; 837\Big\}$