The digits $1,2,3,4$ and 5 were used, each one only once, to write a 5-digit number $a b c d e$, such that: $a b c$ is divisible by $4, b c d$ by 5, and $c d e$ by 3. Find this number.
For $a b c$ to be divisible by 4, its last two digits must form a number divisible by 4. Since the digits are 1, 2, 3, 4, and 5, the only possibilities are: $b c=12, b c=24, b c=32, b c=52$. On the other hand, numbers divisible by 5 end in 0 or 5. Since 0 is not included, it follows that $d=5$ because $b c d$ is divisible by 5. This excludes the possibility $b c=52$ because we cannot repeat the 5. So far, we have 3 possibilities:

$$
a 125 \quad a 245 \quad a 325
$$

Let's now examine these 3 cases to choose the digits $a$ and $e$, remembering that there cannot be any repetition.
![](https://cdn.mathpix.com/cropped/2024_05_01_dbce7bc2a200e234c3e2g-16.jpg?height=482&width=1410&top_left_y=1842&top_left_x=324)

Therefore, the number is 12453.