Interpreting congruence identities as statements about redundancy

Let’s look at the most basic congruence identity, the addition identity:

$$a \equiv b \text{ and } c\equiv d \implies a+c\equiv b+d \pmod m$$

To me, it isn’t immediately obvious how this relates to the typical use of mod I’ve seen, which is computing solutions without integer overflow, and using the Chinese Remainder Theorem to reconstruct the true solution. For example, a typical (simplified) use of congruence looks something like this:

ans = 0
ans = (ans + a) % m
ans = (ans + b) % m
...
ans = (ans + z) % m

print(ans)

This avoid integer overflow, whereas print((a + b + ... + z) % m) does not. Thus when I’ve used congruence identities, I’ve thought of it as a way to insert additional mod operations, which is more obvious in the following statement:

$$a\ \mathrm{mod}\ m + c\ \mathrm{mod}\ m \equiv a+c \pmod m$$

The more general addition identity is useful for proofs, but I think this “where can I add mod operations without changing the answer” interpretation provides useful intuition.