Системи сравнения

[Гост]

Да се решат следните системи сравнения

Система
x[tex]\equiv[/tex]15 (mod 4)
x[tex]\equiv[/tex]9 (mod 7)
x[tex]\equiv[/tex]40 (mod 9)

Система
x[tex]\equiv[/tex]2 (mod 11)
x[tex]\equiv[/tex]5 (mod 15)
x[tex]\equiv[/tex]3 (mod 9)
x[tex]\equiv[/tex]5 (mod 7)

[ammornil]

$ \begin{array}{|l} x \equiv \hspace{0.2em} 2 (mod \hspace{0.2em} 11) \\ x \equiv \hspace{0.2em} 5 (mod \hspace{0.2em} 15) \\ x \equiv \hspace{0.2em} 3 (mod \hspace{0.2em} 9) \\ x \equiv \hspace{0.2em} 5 (mod \hspace{0.2em} 7) \end{array} \\[12pt] \begin{array}{|l} x = 11k + 2 \\ x = 15m +5 \\ x=9n+3 \\ x= 7p+5 \end{array}, \quad \{k, m , n , p \} \in\mathbb{R}, \quad \Leftrightarrow \begin{array}{|l} x = 15m +5 \\ x= 7p+5 \\ x = 11k + 2 \\ x=9n+3 \end{array}\\[6pt] \because{} 15m+5 =7p+5 \Rightarrow 15m =7p \Rightarrow p=2m +\dfrac{m}{7} \\[6pt] \quad \dfrac{m}{7}=a \in{\mathbb{R}} \Rightarrow m=7a, \Rightarrow p=15a \Rightarrow x=7\cdot{15a}+5= 105a +5 \\[6pt] \quad \boxed{x=105a +5} \\[12pt] \because{} 105a +5= 11k +2 \Rightarrow k= 9a +3 +\dfrac{6a}{11} \\[6pt] \quad \dfrac{6a}{11}= b \in\mathbb{R} \Rightarrow a= b +\dfrac{5b}{6} \\[6pt] \quad \dfrac{5b}{6}= c\in{\mathbb{R}} \Rightarrow b= c +\dfrac{c}{5} \\[6pt] \quad \dfrac{c}{5}= d \in{\mathbb{R}} \Rightarrow c=5d \Rightarrow b=6d \Rightarrow a=11d \Rightarrow k= 9\cdot{11d} +3 +6d= 105d +3 \\[6pt] \quad x= 11\cdot{(105d+3)} +3= 1155d +36 \\[6pt] \quad \boxed{x=1155d +36} \\[12pt] \because{} 1155d +36= 9n +3 \hspace{0.2em}|\div{3} \Leftrightarrow 385d +12= 3n +1 \Rightarrow n= 128d +11 +\dfrac{d}{3} \\[6pt] \quad \dfrac{d}{3}= t \in{\mathbb{R}} \Rightarrow d=3t \Rightarrow n= 128\cdot{3t}+11+t =385t +11 \\[6pt] \quad x=9\cdot{(385t +11)} +3= 3465t +102 \Leftrightarrow x\equiv \hspace{0.2em} 102 (mod \hspace{0.2em} 3465) \\[12pt] $ $$ \text{Отг.}\quad x\equiv \hspace{0.2em} 102 (mod \hspace{0.2em} 3465) $$Прегледайте сметките за грешки.

[ammornil]

В горното съм допуснал фактическа грешка в полето на променливите, вместо "$\in{\mathbb{R}}$" във всички случаи следва да бъде "$\in{\mathbb{N}}$". Моля да ме извините за грешката.