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Auto: K Kovandzhei A.K.

A student did not notice the multiplication sign between two seven-digit numbers and wrote a single fourteen-digit number, which turned out to be three times their product. Find these numbers.
Let $x$ and $y$ be the required numbers. According to the condition, $3xy = 10^7 x + y$.

First method. $3y = 10^7 + y/x$. Since $0 < y/x < 10$, then $10^7 < 3y < 10^7 + 10$. Therefore, $33333331/3 < y < 33333362/3$. Hence, $y' < 4$, that is,

$10^7 + 1 \leq 3y < 10^7 + 4$. Only one number in this interval is divisible by 3: this is $10^7 + 2$. Therefore, $y = 3333334, x = 1666667$.

Second method. $10^7 x = (3x - 1)y$. Since $x$ and $3x - 1$ are coprime, then $3x - 1$ is a divisor of $10^7$. But $3x - 1 \geq 3 \cdot 10^6 - 1$, so $3x - 1 = 5 \cdot 10^6$. From this,

$x = 1666667, y = 3333334$.

## Answer

1666667 and 3333334.

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