Our method for solving xk ≡ b (mod m) is first to find integers u and v satisfying ku − φ(m)v = 1, and then the solution is x ≡ bu (mod m). However, we only showed that this works provided that gcd(b,m) = 1, since we used Euler’s formula bφ(m) ≡ 1 (mod m).
(a) If m is a product of distinct primes, show that x ≡ bu (mod m) is always a solution to xk ≡ b (mod m), even if gcd(b,m) > 1.
(b) Show that our method does not work for the congruence x5 ≡ 6 (mod 9).
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