the watered-down version of the crt (which is all you need for stuff like this) tells us that there is a natural map from

to

, for distinct primes

and positive integers

, and it is an isomorphism.

To apply the theorem here, we have an element

, and we want to find the associated element in

. By the euclidean algorithm we can find linear combinations

each of these equalities corresponds to a value 1, 2 or 3 (mod 8, 25 and 81, respectively). Take the first term in the left of each expression, multiply it by the corresponding value 1/2/3, and then add them all together. We get:

and this is the solution mod

. Reducing we get

.

This equivalence class captures all the solutions to the system by the crt.