Is there any known algorithm for computing the GCD of two numbers in the modular representation (i.e, residues modulo pair-wise coprime integers) that does not require the computation of the actual integers using Chinese Remainder Theorem?
Let the modulii set $M = \{2, 3, 5, 7\}$. The residue representation of $x$ is written as $x \bmod M$ given by
$$ x \bmod M = \langle x \bmod 2, x \bmod 3, x \bmod 5, x \bmod 7 \rangle \mod M$$
$$24 = \langle 0, 0, 4, 3 \rangle \mod M$$ $$36 = \langle 0, 0, 1, 1 \rangle \mod M$$
We know $GCD(24, 36) = 12 = \langle 0, 0, 2, 1\rangle \mod M$.
In general, is there a way to compute the modular representation of the GCD directly in the residue form without converting the respective residue representations into the integers {$(24, 36)$ in this example}, and without using the traditional integer GCD computation?
If there is a previously known method, please provide a reference to the text or paper.
Related:
Arithmetic inequality comparison of integers in residues modulo primes