Question

Suma a trei numere naturale este 435. Impartind aceste numere la 6 , 7 , 8 pbtinem aceleasi caturi si acrleasi resturi. Determinati cele trei numere. DAU COROANA

Answer 1

Notăm cele trei numere cu a,  b,  c.

$\it a:6 = x\ rest \ r \ \Rightarrow\ \begin{cases} \it a = 6x + r \\ \\\it r < 6\Rightarrow r\in \{0,1,2,3,4,5\} \end{cases}$

$\it b:7=x\ rest\ r \Rightarrow b=7x+r \\\\c:8=x\ rest\ r \Rightarrow c=8x+ r \\\\a+b+c=435 \Rightarrow 6x+r+7x+r+8x+r=435 \Rightarrow 21x+3x=435|_{:3} \Rightarrow \\\\\Rightarrow 7x+r=145 \Rightarrow 7x+r=140+5 \Rightarrow 7x+r=7\cdot20+5 \Rightarrow x=20,\ r=5$

$\it a=6x+r=6\cdot20+5=125\\\\b=7x+r=7\cdot20+5=145\\\\c=8x+r=8\cdot20+5=165$