Răspuns: 28 este cel mai mic număr ce respecta conditiile problemei
Notam cu n numarul căutat
n : 6 = c₁ rest 4 ⇒ n = 6 • c₁ + 4 | +2
n : 5 = c₂ rest 3 ⇒ n = 5 • c₂ + 3 | +2
n + 2= 6c₁ + 6 ⇒ n + 2 = 6(c₁ + 1)
n + 2 = 5c₂ + 5 ⇒ n + 2 = 5(c₂ + 1)
n + 2 = 6 · 5
n + 2 = 30
n = 30 - 2
n = 28
Verificare:
28 : 6 = 4 rest 4 (adevarat)
28 : 5 = 5 rest 3 (adevarat)
Notăm numărul cerut cu n.
[tex]\it \left.\begin{aligned}\ \it n:6=a\ \ rest\ 4 \Rightarrow n=6a+4\\ \\ \it n:5=b\ \ rest\ 3 \Rightarrow n=5b+3 \end{aligned}\right\} \Rightarrow 5b+3=6a+4 \Rightarrow b=\dfrac{6a+1}{5} \Rightarrow \\ \\ \\ \Rightarrow b=\dfrac{5a+a+1}{5}=\dfrac{5a}{5}+\dfrac{a+1}{5}=a+\dfrac{a+1}{5}\\ \\ \\ b_{min} = 4+1=5 \Rightarrow n_{min}=5\cdot5+3=28[/tex]
Sau:
[tex]\it \left.\begin{aligned}\ \it n:6=a\ \ rest\ 4 \Rightarrow n=6a+4|_{+2} \Rightarrow n+2\in M_6 \\ \\ \it n:5=b\ \ rest\ 3 \Rightarrow n=5b+3|_{+2} \Rightarrow n+2\in M_5 \end{aligned}\right\} \Rightarrow\\ \\ \\ \Rightarrow n+2=(6,\ 5 ) \Rightarrow n+2=30|_{-2} \Rightarrow n=28 \ \ (valoarea\ \ minim\breve a)[/tex]