-123 = a(mod 10)

Step 1 (Modulo Definition): The congruence $-123\equiv a(\mathrm{mod}10)$ means that:

where $k$ is an integer and $a$ is the unique remainder in the standard residue range $[0,9]$.

Step 2 (Solve for a): We find the largest multiple of 10 that is less than or equal to $-123$, which is $-130$:

This matches the form with $k=-13$ and $a=7$.

Step 3 (Conclusion): The value of $a$ is exactly 7.