In this chapter we described how to compute a kth root of b modulo m, but you may well have asked yourself if b can have more than one kth root. Indeed it can! For example, if a is a square root of b modulo m, then clearly −a is also a square root of b modulo m.
(a) Let b, k, and m be integers that satisfy
gcd(b,m) = 1 and gcd( k, φ(m)) = 1.
Show that b has exactly one kth root modulo m.
(b) Suppose instead that gcd ( k, φ(m) )> 1. Show that either b has no kth roots modulo m, or else it has at least two kth roots modulo m. (This is a hard problem with the material that we have done up to this point.)
(c) If m = p is prime, look at some examples and try to find a formula for the number of kth roots of b modulo p (assuming that it has at least one).
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