3. Find the smallest trimestral number for which all the digits of its triple are even.
I/3. 1. way Let's denote the three-digit number as $\overline{a b c}$. The triple of this number is

$$
3 \cdot \overline{a b c}=(3 a) \cdot 100+(3 b) \cdot 10+3 c
$$

The smallest possible digit in the hundreds place is $a=1$. For the digit in the hundreds place of $3 \cdot \overline{a b c}$ to be even, it must be that $3 b \cdot 10+3 c \geq 100$, from which it follows that $10 b+c \geq \frac{100}{3}=33+\frac{1}{3}$. The smallest number that satisfies this inequality is 34, so the desired number is 134, and its triple is 402.

![](https://cdn.mathpix.com/cropped/2024_06_07_24964e3bc413fb5af60fg-07.jpg?height=206&width=1650&top_left_y=591&top_left_x=220)

2. way Let's consider what a number composed entirely of even digits, which is a multiple of a natural number, could be. Since the number is a three-digit number, its triple is at least 300. Since the triple has only even digits, the digit in the hundreds place is at least 4. The smallest three-digit number that is at least 400 and divisible by 3 is 402. All digits are even, so $\frac{402}{3}=134$ is the desired three-digit number.

The realization that the triple is at least 300 ..................................................... 2 points

The realization that the digit in the hundreds place of the triple is at least $4 \ldots \ldots \ldots \ldots \ldots \ldots . \ldots .$.

![](https://cdn.mathpix.com/cropped/2024_06_07_24964e3bc413fb5af60fg-07.jpg?height=60&width=1633&top_left_y=1192&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_24964e3bc413fb5af60fg-07.jpg?height=55&width=1633&top_left_y=1246&top_left_x=220)