Number of Five Digit Numbers Divisible by Four From Specific Digits

To find the total number of $5$-digit numbers formed using the digits from the set $S = \{1, 2, 3, 4, 5\}$ (with repetition allowed) that are divisible by $4$, we use the divisibility rule for $4$.

A number is divisible by $4$ if and only if the number formed by its last two digits is divisible by $4$.

Let the $5$-digit number be represented as $d_5 d_4 d_3 d_2 d_1$, where $d_i \in \{1, 2, 3, 4, 5\}$ for $i = 1, 2, 3, 4, 5$.

Step 1: Count the possible choices for the last two digits ($d_2 d_1$)

We need the two-digit number $d_2 d_1$ to be a multiple of $4$ using digits from $\{1, 2, 3, 4, 5\}$:

• If $d_2 = 1$, the possible two-digit number is $12$ (divisible by $4$).
• If $d_2 = 2$, the possible two-digit number is $24$ (divisible by $4$).
• If $d_2 = 3$, the possible two-digit number is $32$ (divisible by $4$).
• If $d_2 = 4$, the possible two-digit number is $44$ (divisible by $4$).
• If $d_2 = 5$, the possible two-digit number is $52$ (divisible by $4$).

Thus, there are $5$ valid pairs for the last two digits $(d_2, d_1) \in \{(1, 2), (2, 4), (3, 2), (4, 4), (5, 2)\}$.

Step 2: Count the possible choices for the remaining digits ($d_5, d_4, d_3$)

Since repetition of digits is allowed, each of the first three positions ($d_5, d_4, d_3$) can be filled by any of the $5$ digits from the set $\{1, 2, 3, 4, 5\}$.

• Number of choices for $d_5 = 5$
• Number of choices for $d_4 = 5$
• Number of choices for $d_3 = 5$

The total number of ways to choose the first three digits is: $5 \times 5 \times 5 = 5^3 = 125$

Step 3: Calculate the total count

By the fundamental principle of counting, the total number of such $5$-digit numbers is: $\text{Total Numbers} = 5^3 \times 5 = 625$