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The multiplicative inverse {{math|''x'' ≡ ''a''<sup>−1</sup> (mod ''m'')}} may be efficiently computed by solving [[Bézout's identity|Bézout's equation]] {{math|1=''a x'' + ''m y'' = 1}} for {{math|''x''}}, {{math|''y''}}, by using the [[Extended Euclidean algorithm]].
In particular, if {{math|''p''}} is a prime number, then {{math|''a''}} is coprime with {{math|''p''}} for every {{math|''a''}} such that {{math|0 < ''a'' < ''p''}}; thus a multiplicative inverse exists for all {{math|''a''}} that is not congruent to zero modulo {{math|''p''}}.
== Advanced properties ==
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