[tex]\it \{a,b,c\}\ d.\ p.\ \{2,3,4\} \Rightarrow \dfrac{a}{2}= \dfrac{b}{3}= \dfrac{c}{4}=k \Rightarrow\begin{cases} \it a=2k\\\it b=3k\\ \it c=4k\end{cases} \ \ \ (1)\\ \\ \\ \{c,d,e\}\ i.\ p.\ \{2,3,4\} \Rightarrow {c,d,e\}\ d.\ p.\ \left\{\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4} \right\} \Rightarrow \begin{cases} \it c=\dfrac{m}{2}\\ \\\it d=\dfrac{m}{3}\\ \\ \it e=\dfrac{m}{4}\end{cases} \ \ \ (2)[/tex]
[tex]\it (1),(2) \Rightarrow c=\dfrac{m}{2}=4k \Rightarrow m=8k\ \ \ \ (3)\\ \\ (2),(3) \Rightarrow d=\dfrac{8k}{3},\ e=\dfrac{8k}{4}=2k\ \ \ \ (4)\\ \\ a+b+c+d+e=123 \stackrel{(1),(4)}{\Longrightarrow}\ 2k+3k+4k+\dfrac{8k}{3}+2k=123 \Rightarrow\\\\ \Rightarrow\ ^{3)}11k+\dfrac{8k}{3}=\ ^{3)}123 \Rightarrow 33k+8k=3\cdot123 \Rightarrow 41k=3\cdot123|_{:41} \Rightarrow\\ \\ \Rightarrow k=3\cdot3\Rightarrow k=9\ \ \[/tex]
Înlocuim k = 9 în relațiile (1),(4) și obținem:
a=2·9 = 18
b= 3·9 = 27
c= 4·9 = 36
d= 8·9/3 = 8·3 = 24
e = 2·9 = 18
