It is easy to see that it works: 49*10= 490 and 490/36= 13 with remainder 22.
As for an "easier way", what you did looks pretty easy to me! Certainly better than multiplying 48 by 1, 2, 3, etc. until you find the right one.
Hello, dwsmith!
Here is a very primitive solution . . .
Solve:. .
The congruence reduces to: .
This means: .
.[1]
Since is an integer, must be divisible by 13.
We have: .
.[2]
Since is an integer, must be divisible by 10.
The first time this happens is when
Substitute into [2]: .
Substitute into [1]: .
Therefore: .