## Task 13/80

From an arbitrarily chosen natural number $n_{1}$, form a natural number $n_{2}$ by appending a three-digit natural number. For which numbers $\mathrm{n}$ is the sum $s=n_{1}+n_{2}$ divisible by 77 without a remainder?
It is $n_{2}=1000 n_{1}+n$, thus $s=n_{1}+n_{2}=1001 n_{1}+n$. Since $1001=7 \cdot 11 \cdot 13$ is divisible by $77=7 \cdot 11$, the divisibility of $s$ is determined precisely when $n$ is divisible by 77.

This yields the following 11 three-digit numbers $\mathrm{n}$:

$$
154 ; 231 ; 308 ; 385 ; 462 ; 539 ; 616 ; 693 ; 770 ; 847 ; 924
$$

If "improperly three-digit" numbers are also allowed, then 077 also satisfies the condition; if the number zero is included in the set of natural numbers, then $n=000$ must also be considered, so that a maximum of 13 numbers $n$ can be listed.