Summary: We present new division algorithms for Residue Number Systems (RNS). The algorithms are based on a sign estimation procedure that computes the sign of a residue number to be positive, negative, or indeterminate. In the last case, magnitude of the number is guaranteed to be in a limited interval whose size is related to the cost of the sign estimation process. Our division algorithms resemble SRT (Sweeney, Robertson, and Tocher) division; quotient digits in the set \(\{-1,0,1\}\) are computed one by one. Assume that the RNS has \(n\) moduli, \(n\) residue processors, and \(b\) bits per modulus, and that each \(b\)-bit addition/subtraction takes unit time. Our sign estimation procedure uses relatively small lookup tables and takes \(O(\log n)\) time. The first division algorithm based on the new sign estimation procedure requires \(O(nb\log n)\) time. A second algorithm, which improves the time complexity to \(O(nb)\), is the fastest algorithm proposed thus far. Intermediate between the two algorithms are a number of choices that offer speed/cost tradeoffs.