[ Processes and operations]

Given a natural number \$N\$. Vera performs the following operations with it: first, she adds 3 until the resulting number is divisible by 5 (if \$N\$ is initially divisible by 5, nothing needs to be added). The resulting number is then divided by 5. She then performs these same operations with the new number, and so on. From which numbers can 1 not be obtained by these operations?
Indeed, the number $1$ cannot be obtained from numbers that are multiples of $3$, because if a number $N$ is a multiple of $3$, then $N + 3$ is also a multiple of $3$, and if $N = 5k$ is a multiple of $3$, then $k$ is a multiple of $3$, since $3$ and all numbers resulting from these operations will be multiples of $3$ but

Let $N$ not be a multiple of $3$ with different remainders when divided by $5$, so one of them is a multiple of $5$. Therefore, after the first division by $5$, Vera will get a number that is not a multiple of $3$ and does not exceed $\frac{N+12}{5} = 0.2N + 2.4$, which is strictly less than $N$ for $N > 12$ until the number $1$ is obtained or

Comment. The formulation of this problem is similar to the famous open problem — the Collatz conjecture. With a given natural number $N$, the following operation is performed: if $N$ is even, it is divided by 2, and if it is odd, it is multiplied by 3 and 1 is added to the result (yielding the number $3N+1$), after which the process is repeated. The conjecture states that eventually, the number 1 will be obtained as a result of these operations. As of now, using distributed computing, the conjecture has been verified up to numbers on the order of $10^{21}$, and in the most complex cases, it takes about 3000 steps to reach 1.

## Answer

$3k, k \in \mathbb{N}$.

Submit a comment