This work primarily investigates the cyclic properties of the remainder sequence of the Repunit sequence =<1, 11, 111,…> modulo n. We explore the conditions under which the remainder sequence of Repunit numbers forms a purely periodic sequence, a pre-periodic sequence, or a perfectly periodic sequence, and we provide the formula and the upper bound for the cyclic period. Furthermore, we discover that the cyclic period of a first-order non-homogeneous linear recurrence sequence modulo n is the same as that of the base-c Repunit sequence modulo n/gcd(n,c). We also explore the conditions under which the remainder sequence forms a purely periodic sequence, a pre-periodic sequence, or a perfectly periodic sequence.