Will the number $3^{444}+4^{333}$ be divisible by 5?
There is a pattern for the unit digit of a power of 3: it has a period of 4, as it repeats every four times. Indeed, we have

$$
\begin{array}{ll}
3 & 3^{5}=243 \\
3^{2}=9 & 3^{6}=\ldots 9 \\
3^{3}=27 & 3^{7}=\ldots 7 \\
3^{4}=81 & 3^{8}=\ldots 1
\end{array}
$$

Since 444 is a multiple of 4, the unit digit of $3^{444}$ is 1.

Analogously, the unit digit of powers of 4 has a period of 2. Indeed, we have

$$
\begin{array}{ll}
4^{1}=4 & 4^{3}=64 \\
4^{2}=16 & 4^{4}=256
\end{array}
$$

Since 333 is odd, the unit digit of $4^{333}$ is 4. Therefore, the unit digit of $3^{444}+4^{333}$ is $1+4=5$, which means it is divisible by 5.

REMINDER: Any number ending in 0 or 5 is divisible by 5.