Question

sa se determine numerele naturale a,b,c cu proprietatea a/3=b/5=7/c

Answer 1

a/3=b/5=7/c

c∈N*

a=3*(7/c)∈N
b=5*(7/c)∈N

deci fie (7/c)∈N, fie c|3 si c|5
dac 7/c∈N atunci c1 sau c=7
pt c=1
a/3=b/5=7⇒a=3*7=21 si b=5*7=35

pt c=7, a =3*(7/7)=3 si b- 5*(7/7)=5



dac c|3 si c|5 atunci c=(3.5)=1 dar acest caz a fost deja studiat

deci solutii
(a;b;c)∈{(21;35;1);(3;5;7)}

Answer 2

[tex]\it \dfrac{a}{3} = \dfrac{b}{5}= \dfrac{7}{c}\ \ \ \ \ (*) \\ \\ \\ (*) \Rightarrow \dfrac{a}{3} = \dfrac{7}{c} \Rightarrow a= \dfrac{21}{c} \in\mathbb{N} \Rightarrow c\in \{1,\ 3,\ 7,\ 21\}\ \ \ \ \ (1) \\ \\ \\ (*) \Rightarrow \dfrac{b}{5} = \dfrac{7}{c} \Rightarrow b= \dfrac{35}{c} \in\mathbb{N} \Rightarrow c\in \{1,\ 5,\ 7,\ 35\}\ \ \ \ \ (2)[/tex]

$\it (1), (2) \Rightarrow c\in \{1,\ 3,\ 7,\ 21\} \cap \{1,\ 5,\ 7, \ 35\} \Rightarrow c \in \{1,\ 7\}$

[tex]\it c=7\stackrel{(*)}{\Longrightarrow} \begin{cases}\it \dfrac{a}{3} =1 \Rightarrow a=3\\ \\ \it \dfrac{b}{5} =1 \Rightarrow b=5 \end{cases} \\ \\ \\ c=1\stackrel{(*)}{\Longrightarrow} \begin{cases}\it \dfrac{a}{3} =7 \Rightarrow a=21\\ \\ \it \dfrac{b}{5} =7 \Rightarrow b=35 \end{cases} [/tex]


$\it (a,\ b,\ c) \in\{(3,\ 5,\ 7),\ (21,\ 35,\ 1)\}$