Exercițiul 102

$43x^2 \lt 1720 \textcolor{red}{\Rightarrow} x^2 \lt 40 \textcolor{red}{\Rightarrow} x \in \{2, 3, 5\}$

$\bold{Caz \space 1}$: $x=2 \textcolor{red}{\Rightarrow} 129 \cdot y \space este \space par \textcolor{red}{\Rightarrow} y=2$
$\hspace*{2em} 43 \cdot 2^2 + 129 \cdot 2 + 10z = 1720$
$\hspace*{2em} 10z=129 \textcolor{red}{\Rightarrow} z=129$ - nu convine

$\bold{Caz \space 2}$: $\boxed{x=3}$
$\hspace*{2em} 43 \cdot 3^2 + 129 \cdot y + 10z = 1720$
$\hspace*{2em} 129y+10z=1333 \textcolor{red}{\Rightarrow} U_c(y)=7$
$\hspace*{2em}\bullet \boxed{y=7} \textcolor{red}{\Rightarrow} 129\cdot 7 + 10z=1333 \textcolor{red}{\Rightarrow} \boxed{z=43}$
$\hspace*{2em}\bullet \space y=17 \textcolor{red}{\Rightarrow} 129\cdot 17 = 2193 > 1333$ - nu convine
$\hspace*{2em}\bullet \space$ orice alt nr. cu $U_c(y)=7$ - nu convine

$\bold{Caz \space 3}$: $x=5 \textcolor{red}{\Rightarrow} U_c(129 \cdot y) = 5 \textcolor{red}{\Rightarrow} y=5$
$\hspace*{2em} 43 \cdot 5^2 + 129 \cdot 5 + 10z = 1720$
$\hspace*{2em} 1720+ 10z = 1720 \textcolor{red}{\Rightarrow} z=0$, nu convine

Deci $x+y+z = 3+7+43 = 53.$