A number less than 200 is formed by three different digits, and the double of this number also has all different digits. Moreover, the number and its double do not have any digits in common. What is this number? How many solutions does this problem have?
Initially note that the double of an integer is even, so it ends in $0,2,4,6$ or 8. However, the number we are looking for cannot end in 0, because in that case its double would also end in 0, and both would have the digit 0 in common. Therefore, we have the following cases.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-087.jpg?height=366&width=240&top_left_y=251&top_left_x=428)
![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-087.jpg?height=356&width=918&top_left_y=256&top_left_x=770)

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-087.jpg?height=363&width=234&top_left_y=641&top_left_x=434)
![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-087.jpg?height=364&width=918&top_left_y=640&top_left_x=772)

Now, let's determine all the possibilities for each case, always remembering that the number and its double cannot have common digits.

- Case I - we have three possibilities:

$$
152 \times 2=304, \quad 182 \times 2=364, \quad 192 \times 2=384
$$

- Case II - we have two possibilities:

$$
143 \times 2=286, \quad 153 \times 2=206
$$

- Case III - we have three possibilities:

$$
134 \times 2=268, \quad 154 \times 2=308, \quad 164 \times 2=328
$$

- Case IV - we have three possibilities:

$$
135 \times 2=270, \quad 145 \times 2=290, \quad 185 \times 2=370
$$

- Case V - we have two possibilities:

$$
176 \times 2=352, \quad 186 \times 2=372
$$

- Case VI - there are no possibilities.
- Case VII - we have three possibilities:

$$
138 \times 2=276, \quad 148 \times 2=296, \quad 178 \times 2=356
$$

- Case VIII - we have two possibilities:

$$
139 \times 2=278, \quad 179 \times 2=358
$$

Thus, we have $3+2+3+2+3+3+2=18$ solutions to this problem, namely: