Exercițiul 101

$4^x \cdot 3^y \space \vdots \space 4 \textcolor{red}{\Rightarrow} \overline{y30x} \space \vdots \space 4 \textcolor{red}{\Rightarrow} \overline{0x} \space \vdots \space 4 \textcolor{red}{\Rightarrow} x \in \{0, 4, 8\}$

$\bold{Caz \space 1}$: $x=0$, nu convine ($U_c(4^x \cdot 3^y) \not = 0$)

$\bold{Caz \space 2}$: $\boxed{x=4} \textcolor{red}{\Rightarrow} 4^x \cdot 3^y = \overline{y304}$
$\hspace*{2em} 4^x \cdot 3^y \space \vdots \space 3 \textcolor{red}{\Rightarrow} \overline{y304} \space \vdots \space 3 \textcolor{red}{\Rightarrow} (y+3+0+4) \space \vdots \space 3 \textcolor{red}{\Rightarrow} y \in \{2, 5, 8\}$
$\hspace*{2em}\bullet \space \boxed{y=2} \textcolor{red}{\Rightarrow} 256 \cdot 3^2=2304$ - adevărat
$\hspace*{2em}\bullet \space y=5 \textcolor{red}{\Rightarrow} 256 \cdot 3^5=5304$ - fals
$\hspace*{2em}\bullet \space y=8 \textcolor{red}{\Rightarrow} 256 \cdot 3^8=8304$ - fals

$\bold{Caz \space 3}$: $x=8$, nu convine ($4^8=2^{16} \gt 9999$)

Deci $x-y=4-2=2.$