The digits 1, 2, 3, 4, and 5 were used, each one only once, to write a certain five-digit number $a b c d e$ such that $a b c$ is divisible by $4, b c d$ is divisible by 5, and $c d$ is divisible 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 $b c=12, b c=24, b c=32$, and $b c=52$. 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 $b c=52$, as we cannot repeat the 5. So far, we have three possibilities, namely,

$$
\text { a125e, } a 245 e \text { and a325e. }
$$

Let's examine these three cases to choose the digits $a$ and $e$, remembering that there should be no repetition.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-085.jpg?height=483&width=462&top_left_y=798&top_left_x=363)

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-085.jpg?height=423&width=297&top_left_y=802&top_left_x=845)

Not a multiple of 3 Is a multiple of 3

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-085.jpg?height=368&width=280&top_left_y=798&top_left_x=1342)

$43251 \quad 13254$

Not a multiple of 3 Not a multiple of 3

Therefore, the number is 12453.