The number of 5 digit numbers which are divisible by 4, with digits from the set $\{1, 2, 3, 4, 5\}$ and the repetition of digits is allowed, is ______.

$\text{Digits allowed: } \{1,2,3,4,5\}, \text{ repetition allowed}$ $\text{Five-digit numbers divisible by } 4$ $\text{To find: the number of such 5-digit numbers.}$ $4 \mid 100x \ \forall\, x \in \mathbb{Z} \Rightarrow 4 \mid N \iff 4 \mid (\text{last two digits of } N)$ $\overline{de} \in \{12,\ 24,\ 32,\ 44,\ 52\} \quad \Rightarrow 5 \text{ ways}$ $12=4(3),\ 24=4(6),\ 32=4(8),\ 44=4(11),\ 52=4(13)$ $5 \times 5 \times 5 = 5^{3} = 125$ $5 \times 125 = 625$ $\dfrac{5^{5}}{5} = 5^{4} = 625 \quad \checkmark$ $\boxed{625}$