# Problem 7.4 (7 points)

Find all three-digit numbers that decrease by 6 times after the first digit is erased. #
# Solution:

Let $\overline{a b c}$ be the desired number. According to the condition, $100 \mathrm{a}+10 \mathrm{~b}+\mathrm{c}=6(10 \mathrm{~b}+\mathrm{c})$, from which we get $100a = 50 \mathrm{~b}+5 \mathrm{c}$, or $20 \mathrm{a}=10 \mathrm{~b}+\mathrm{c}$. Therefore, $\mathrm{c}=10(2 \mathrm{a}-\mathrm{b})$. Since $\mathrm{c}$ is a single-digit number, $\mathrm{c}=0$. Thus, $\mathrm{b}=2 \mathrm{a}$. From this, we get the numbers 120, 240, 360, 480. There are no other numbers because $\mathrm{b}<10$.

| Criteria | Points |
| :--- | :---: |
| Complete solution | 7 |
| All numbers found, without proof that there are no other numbers | 4 |
| Several numbers found | 2 |
| The last page number is incorrectly determined | 0 |

Answer: 120, 240, 360, 480.