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.