5. Find all natural numbers $N<10000$ such that $N=26 \cdot S(N)$, where $S(N)$ denotes the sum of the digits of the number $N$.
Answer: 234, 468.

Solution. Note that the number $N$ is at most a three-digit number, otherwise $N \geq 1000 > 936 = 26 \cdot 4 \cdot 9 \geq 26 \cdot S(N)$. It remains to consider numbers less than 1000. As is known, a number and the sum of its digits give the same remainder when divided by 9. Consider the equation $N=26 \cdot S(N)$ from the perspective of divisibility by 9. It is more convenient to rewrite it as: $N-S(N)=25 \cdot S(N)$. Then the left side is divisible by 9, and therefore the right side is also divisible by 9. The numbers 25 and 9 are coprime, so $S(N)$ is divisible by 9, and therefore $N$ is divisible by 9. According to the condition, $N$ is divisible by 26. Thus, $N$ is divisible by $9 \cdot 26 = 234$. It remains to check the numbers that are multiples of 234 and less than 1000.

Comment. It is proven that the number is at most a three-digit number - 2 points. It is proven that the number is divisible by 9 - 2 points. 1 point for each correct answer. Points are cumulative. If the divisibility by 9 is not proven, and only numbers less than 1000 and divisible by 26 are enumerated, then 2 points are given only if the enumeration is complete and recorded.