Problem 4. Find the natural number $\overline{abc}$ that satisfies the condition $\overline{abc}+\overline{bc}+c=687$
Solution. The unit digit of the number $\overline{a b c}+\overline{b c}+c$ is the unit digit of $c+c+c$, that is, $c+c+c$ ends in 7. The only possibility is $c=9$. From this, we get that $100 a+20 b+27=687$, or $5 a+b=33$. The last equation is satisfied if $a=5$ and $b=8$ or $a=6$ and $b=3$. Therefore, the required numbers are 589 and 639.