Modulo Properties Calculator: Associative, Commutative, Distributive

Your details

Any integer. Negative values are handled correctly.
Any integer.
Used in associativity and distributivity checks.
Must be a positive integer >= 2. All residues are in {0, 1, ..., n-1}.
Associative (addition)All properties verified
Yes - holds

Whether the associative law holds for addition mod n

x mod n7
y mod n5
z mod n3
(x + y) + z (mod n)4
x + (y + z) (mod n)4
(x * y) * z (mod n)6
x * (y * z) (mod n)6
Associative (multiplication)Yes - holds
x + y (mod n)1
y + x (mod n)1
Commutative (addition)Yes - holds
x * y (mod n)2
y * x (mod n)2
Commutative (multiplication)Yes - holds
x * (y + z) (mod n)1
(x*y + x*z) (mod n)1
Distributive (mul over add)Yes - holds
All properties verifiedAll 5 properties hold
Assoc Add LHS4
Assoc Add RHS4
Comm Add LHS1
Comm Add RHS1
Dist LHS1
Dist RHS1

All five modular arithmetic properties hold for n = 11.

  • With modulus n = 11, the integers mod 11 form a commutative ring under addition and multiplication.
  • Commutativity: x + y = 1 and y + x = 1 (mod 11) - equal.
  • Associativity of addition: (x+y)+z = 4 and x+(y+z) = 4 (mod 11) - equal.
  • Distributivity: x*(y+z) = 1 and x*y+x*z = 1 (mod 11) - equal.

Next stepThese properties hold for every non-zero modulus and every choice of integers. Try a prime modulus such as 7 or 13: the integers mod p form a field, adding even more structure (every non-zero element has a multiplicative inverse).

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