We define similar numbers as positive integers that have exactly the same digits (but possibly in another order). For example, $1241$, $2114$ and $4211$ are similar numbers, but $1424$ is not similar to the other three. Determine whether there exist three similar numbers, each with $300$ digits (all digits being non-zero), such that the sum of two of them equals the third. If the answer is yes, provide an example; if not, justify why it is impossible.
Problem
Source: 2019 Argentina L2 P4
Tags: number theory
BR1F1SZ
21.01.2025 02:35
jasperE3 wrote:
Its impossible $\pmod{10}$.
Actually, $$\underbrace{459\cdots459}_{\text{300 digits}}+\underbrace{495\cdots495}_{\text{300 digits}}=\underbrace{954\cdots954}_{\text{300 digits}}.$$
jasperE3
21.01.2025 06:20
BR1F1SZ wrote: jasperE3 wrote:
Its impossible $\pmod{10}$.
Actually, $$\underbrace{459\cdots459}_{\text{300 digits}}+\underbrace{495\cdots495}_{\text{300 digits}}=\underbrace{954\cdots954}_{\text{300 digits}}.$$ But $954\ldots954+495\ldots495\ne459\ldots459$. edit oh maybe the problem is badly worded
BR1F1SZ
21.01.2025 18:29
Quote: edit oh maybe the problem is badly worded Yes, Google Translate isn't very good Edit: I fixed it.