Translation Not Available Yet
Modular Arithmetic Properties Calculator is not yet available in Français. Showing the English version below. All formulas and calculations work the same.
Dernière mise à jour : 31 juillet 2026
Modular Arithmetic Properties Calculator
Créateurs
Dharmendra SinghRéviseurs
Créateurs
Dharmendra SinghRéviseurs
Créateurs
Dharmendra SinghRéviseurs
Créateurs
Dharmendra SinghRéviseurs
Formule
((a + b) + c) mod m = (a + (b + c)) mod m; (a + b) mod m = (b + a) mod m; ((ab)c) mod m = (a(bc)) mod m; a(b + c) mod m = (ab + ac) mod m
Où :
- a=First non-negative integer
- b=Second non-negative integer
- c=Third non-negative integer
- m=Positive modulus
Exemples résolus
Verifying all properties mod 5
Use a=4, b=3, c=2, and m=5 to see every property return matching sides.
- 1Compute both associative addition sides modulo 5.
- 2Compute both multiplication sides modulo 5.
- 3Compare the distributive pair and check each boolean flag.
Another associative addition check
Try a=7, b=5, c=3, m=4 to confirm the grouped sums still agree.
- 1Add the first two values before reducing modulo 4.
- 2Add the last two values before reducing modulo 4.
- 3Observe that both grouped sums end at the same remainder.
Rounded integer inputs
Decimal entries are rounded before evaluation, so 4.4 behaves like 4 and 2.6 behaves like 3.
- 1Round inputs to a=4, b=3, c=1, m=6.
- 2Evaluate the modular addition and multiplication pairs.
- 3Read the matching left and right sides.
Invalid modulus example
A modulus of 0 is not allowed because modular arithmetic needs a positive modulus.
- 1Check the modulus before any arithmetic.
- 2Reject zero because m must be at least 1.
- 3Return safe default outputs with an error message.
Introduction
This calculator demonstrates that modular addition and modular multiplication preserve the same core algebraic laws you use with ordinary arithmetic. Enter non-negative integers and a positive modulus to compare both sides of each identity and see the matching remainders.
What This Calculator Verifies
The tool evaluates five classic modular arithmetic properties side by side so you can see the equal remainders directly.
Associativity of addition
Commutativity of addition
Associativity of multiplication
Commutativity of multiplication
Distributivity of multiplication over addition
Associative Addition Modulo m
Grouping does not change the remainder when you add integers modulo m. The calculator reports both grouped sums so the equality is visible instead of assumed.
Left side uses ((a+b)+c) mod m
Right side uses (a+(b+c)) mod m
Matching outputs confirm associativity
Commutative Laws
Swapping the order of addends or factors does not change the final modular result for non-negative integers.
(a+b) mod m equals (b+a) mod m
(a×b) mod m equals (b×a) mod m
The calculator always marks these properties as true
If the numeric left and right sides match, the true flag simply summarizes the evidence.
Distributive Law in Modular Arithmetic
Distributivity survives reduction modulo m, which is why modular arithmetic works so well in algebra, number theory, and cryptography.
Left side computes a×(b+c) mod m
Right side computes (ab+ac) mod m
Equal remainders verify the distributive rule
Input and Validation Rules
All four entries must be finite numbers. The calculator rounds them to the nearest integer, then enforces the non-negative and positive modulus constraints.
a, b, and c must be at least 0 after entry
m must be at least 1
Invalid inputs return zeroed numeric outputs and an error string
Negative values are rejected even if rounding would otherwise move them toward zero.
How to Use the Results
Read each left-hand and right-hand remainder pair first, then use the corresponding true label as a quick conclusion.
Enter a, b, c, and m
Compare each pair of outputs
Use the true flags as a compact summary
Try different moduli to build intuition
Why These Properties Matter
These identities let you regroup, reorder, and expand modular expressions confidently, which simplifies proofs and algorithm design.
Helps with mental math and homework checks
Supports congruence proofs
Useful in modular equations and cryptography
Builds intuition for equivalence classes
FAQ
Do the listed modular properties ever fail?
Not for integer arithmetic with a positive modulus. The calculator is designed to show the matching sides numerically.
Why are the property outputs strings instead of booleans?
The calculator exposes human-readable result fields for UI display, so the verification labels are returned as the strings "true".
Can I enter decimal values?
Yes. The calculator rounds each entry to the nearest integer before applying the modular formulas.
Why must the modulus be positive?
Modulo 0 is undefined, and negative moduli are outside the intended scope of this educational calculator.
Why are a, b, and c restricted to non-negative integers?
The current implementation focuses on the standard classroom setting described in the tool and keeps remainders straightforward to interpret.
What should I compare first?
Start with each left-versus-right numeric pair. The equality of those values is the direct proof for each property.
Is JavaScript modulo safe here?
Yes. Because the calculator uses rounded non-negative integers, JavaScript’s remainder operator behaves exactly as expected for these examples.