Editing Modulo (section)

==Properties (identities)==
{{see also|Modular arithmetic#Properties}}
Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in [[cryptography]] proofs, such as the [[Diffie–Hellman key exchange]]. The properties involving multiplication, division, and exponentiation generally require that {{math|''a''}} and {{math|''n''}} are integers.
* Identity:
** {{math|(''a'' mod ''n'') mod ''n'' {{=}} ''a'' mod ''n''}}.
** {{math|''n''{{sup|''x''}} mod ''n'' {{=}} 0}} for all positive integer values of {{math|''x''}}.
** If {{math|''p''}} is a [[prime number]] which is not a [[divisor]] of {{math|''b''}}, then {{math|''ab''{{i sup|''p''−1}} mod ''p'' {{=}} ''a'' mod ''p''}}, due to [[Fermat's little theorem]].
* Inverse:
** {{math|[(−''a'' mod ''n'') + (''a'' mod ''n'')] mod ''n'' {{=}} 0}}.
** {{math|''b''{{sup|−1}} mod ''n''}} denotes the [[modular multiplicative inverse]], which is defined [[if and only if]] {{math|''b''}} and {{math|''n''}} are [[relatively prime]], which is the case when the left hand side is defined: {{math|[(''b''{{sup|−1}} mod ''n'')(''b'' mod ''n'')] mod ''n'' {{=}} 1}}.
* Distributive:
** {{math|(''a'' + ''b'') mod ''n'' {{=}} [(''a'' mod ''n'') + (''b'' mod ''n'')] mod ''n''}}.
** {{math|''ab'' mod ''n'' {{=}} [(''a'' mod ''n'')(''b'' mod ''n'')] mod ''n''}}.
* Division (definition):  {{math|{{sfrac|''a''|''b''}} mod ''n'' {{=}} [(''a'' mod ''n'')(''b''{{sup|−1}} mod ''n'')] mod ''n''}}, when the right hand side is defined (that is when {{math|''b''}} and {{math|''n''}} are [[coprime]]), and undefined otherwise.
* Inverse multiplication:  {{math|[(''ab'' mod ''n'')(''b''{{sup|−1}} mod ''n'')] mod ''n'' {{=}} ''a'' mod ''n''}}.