properties_congruences - Chapter 18 More Properties of...

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Chapter 18 More Properties of Congruences Theorem 18.1. Let m 2 . If a and m are relatively prime, there exists a unique integer a such that aa 1 (mod m ) and 0 < a < m . We call a the inverse of a modulo m . Note that we do not denote a by a 1 since this might cause some confusion. Of course, if c a (mod m ) then ac 1 (mod m ) so a is not unique unless we specify that 0 < a < m . Proof. If gcd( a, m ) = 1, then by Bezout’s Lemma there exist s and t such that as + mt = 1 . Hence as 1 = m ( t ) , that is, m | as 1 and so as 1 (mod m ). Let a = s mod m . Then a s (mod m ) so aa 1 (mod m ) and clearly 0 < a < m . To show uniqueness assume that ac 1 (mod m ) and 0 < c < m . Then ac aa (mod m ). So if we multiply both sides of this congruence on the left by c and use the fact that ca 1 (mod m ) we obtain c a (mod m ). It follows from Exercise 15.5 that c = a . Remark 18.1 . From the above proof we see that Blankinship’s Method may be used to compute the inverse of a when it exists, but for small m we may 71
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