Editing Modulo (section)

===Modulo with offset===

Sometimes it is useful for the result of {{mvar|a}} modulo {{mvar|n}} to lie not between 0 and {{math|''n'' − 1}}, but between some number {{mvar|d}} and {{math|''d'' + ''n'' − 1}}. In that case, {{mvar|d}} is called an ''offset'' and {{math|1=''d'' = 1}} is particularly common. 

There does not seem to be a standard notation for this operation, so let us tentatively use {{math|''a'' mod<sub>''d''</sub> ''n''}}. We thus have the following definition:<ref name="Mathematica Mod" >{{cite web |url=https://reference.wolfram.com/language/ref/Mod.html |title=Mod |author=<!--Not stated--> |date= 2020  |website= Wolfram Language & System Documentation Center  |publisher=[[Wolfram Research]] |access-date=April 8, 2020 }}</ref> {{math|1=''x'' = ''a'' mod<sub>''d''</sub> ''n''}} just in case {{math|''d'' ≤ ''x'' ≤ ''d'' + ''n'' − 1}} and {{math|1=''x'' mod ''n'' = ''a'' mod ''n''}}. Clearly, the usual modulo operation corresponds to zero offset: {{math|1=''a'' mod ''n'' = ''a'' mod<sub>0</sub> ''n''}}. 

The operation of modulo with offset is related to the [[Floor and ceiling functions|floor function]] as follows:
::<math>a \operatorname{mod}_d n = a - n \left\lfloor\frac{a-d}{n}\right\rfloor.</math>
To see this, let <math display="inline">x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor</math>. We first show that {{math|1=''x'' mod ''n'' = ''a'' mod ''n''}}. It is in general true that {{math|1=(''a'' + ''bn'') mod ''n'' = ''a'' mod ''n''}} for all integers {{mvar|b}}; thus, this is true also in the particular case when <math display="inline">b = -\!\left\lfloor\frac{a-d}{n}\right\rfloor</math>; but that means that <math display="inline">x \bmod n = \left(a - n \left\lfloor\frac{a-d}{n}\right\rfloor\right)\! \bmod n = a \bmod n</math>, which is what we wanted to prove. It remains to be shown that {{math|''d'' ≤ ''x'' ≤ ''d'' + ''n'' − 1}}. Let {{mvar|k}} and {{mvar|r}} be the integers such that {{math|1=''a'' − ''d'' = ''kn'' + ''r''}} with {{math|0 ≤ ''r'' ≤ ''n'' − 1}} (see [[Euclidean division]]). Then <math display="inline">\left\lfloor\frac{a-d}{n}\right\rfloor = k</math>, thus <math display="inline">x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor = a - n k = d +r</math>. Now take {{math|0 ≤ ''r'' ≤ ''n'' − 1}} and add {{mvar|d}} to both sides, obtaining {{math|''d'' ≤ ''d'' + ''r'' ≤ ''d'' + ''n'' − 1}}. But we've seen that {{math|1=''x'' = ''d'' + ''r''}}, so we are done.

The modulo with offset {{math|''a'' mod<sub>''d''</sub> ''n''}} is implemented in [[Mathematica]] as {{code|Mod[a, n, d]}}&thinsp;.<ref name="Mathematica Mod" />