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Mod

Mod[m, n]
gives the remainder on division of m by n.
Mod[m, n, d]
uses an offset d.
  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • For integers m and n, Mod[m, n] lies between 0 and n-1.
  • Mod[m, n, 1] gives a result in the range 1 to n, suitable for use in functions such as Part.  »
  • Mod[m, n, d] gives a result x such that d<=x<d+n and xmodn=mmodn.
  • The sign of Mod[m, n] is always the same as the sign of n, at least so long as m and n are both real.
  • The arguments of Mod can be any numeric quantities, not necessarily integers.  »
  • For positive x, Mod[x, 1] gives the fractional part of x.
  • For exact numeric quantities, Mod internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
  • Mod works with complex numbers, using its definition in terms of Quotient.
  • Mod automatically threads over lists.
EXAMPLESCLOSE ALL
Basic Examples   (4)
Remainders mod 3:
Remainders mod 3 offset to start with 1:
Plot the sequence with fixed modulus:
Plot the sequence, varying the modulus:
Remainders mod 3:
In[1]:=
Click for copyable input
Out[1]=
 
Remainders mod 3 offset to start with 1:
In[1]:=
Click for copyable input
Out[1]=
 
Plot the sequence with fixed modulus:
In[1]:=
Click for copyable input
Out[1]=
 
Plot the sequence, varying the modulus:
In[1]:=
Click for copyable input
Out[1]=
Scope   (8)
Reduce an approximate number:
Reduce an exact numeric quantity:
Reduce mod pi:
Complex number:
Wilson's theorem:
Solve a modular equation:
Evaluate an integral:
Applications   (4)
Extract parts cyclically:
Select primes below 100 having the form of 6 k+1:
Simulate a particle bouncing in a non-commensurate box:
Define a notation for addition mod 2:
Properties & Relations   (2)
The results have the same sign as the second argument:
Possible Issues   (2)
Expressions generated by Mod can be difficult to evaluate with machine precision:
Machine-precision numerical evaluation gives the wrong answer:
Arbitrary precision gives the correct answer:
Some computations may require higher internal precision than the default:
Neat Examples   (3)
Binomial coefficients mod 2:
Mod 4 additive cellular automaton:
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