3+ [ Divisibility of numbers. General properties ]

Humpty-Dumpty walks along a straight line, either taking 37 steps to the left or 47 steps to the right in a minute.

What is the shortest time in which he can be one step to the right of the starting point?
In fact, it is required to find a solution to the equation $47 x-37 y=1 \quad(*)$ in non-negative integers with the smallest sum $x+y$.

One of the solutions to this equation is $-x=26, y=33$. Let $(x, y)$ be another integer solution to this equation. Subtracting the equation $47 \cdot 26-37 \cdot 33=1$ from (*), we get $47(x-26)=37(y-33)$. From this, it follows that $x-26$ is divisible by 37, and $y-33$ is divisible by 47. Therefore, (26, 33) is the solution with the smallest positive value for both $x$ and $y$.

## Otvet

In 59 minutes.