Fix integers b ≥ 2 and k ≥ 1. Define the sequence {z_n} recursively by taking z₀ to be any integer, and for n ≥ 1, taking z_n to be the least nonnegative residue of bz_n-1 modulo (n+k). Since the modulus increases by 1 when stepping from one term to the next, such a definition will be called a running modulus recursion or rumor for short. While the terms of such sequences appear to bounce around irregularly, empirical evidence suggests the terms will eventually be zero. We prove this is so when one additional assumption is made, and we conjecture that this additional condition is always met.