Modulo Calculator (Mod Rechner)
Calculate remainders, quotient, and congruence in one click.
Tip: Use whole numbers only (positive or negative). Large integers are supported.
What is modulo?
Modulo (often written as mod) gives the remainder after division. If you divide a by n, then a mod n is what is left over. For example, 29 mod 7 = 1, because 29 = 4×7 + 1.
A mod rechner is simply a modulo calculator. It helps you compute:
- The remainder of division
- The quotient (how many whole times n fits into a)
- Whether two numbers are congruent modulo n
How to use this mod rechner
Step 1: Enter a
Put your main number in the Number a field. This is the value you want to reduce modulo n.
Step 2: Enter n
Enter the modulus in Modulus n. This cannot be zero. If you enter a negative modulus, the calculator automatically uses its absolute value for the Euclidean remainder.
Step 3 (optional): Enter b
If you also enter Number b, the tool checks congruence: it tells you whether a ≡ b (mod n).
Why modulo matters in real life
Modulo arithmetic is everywhere, even if you do not notice it:
- Time calculations: clocks run mod 12 or mod 24.
- Programming: cycling array indices, hash tables, and wrap-around counters.
- Cryptography: modular arithmetic powers RSA and many security protocols.
- Number theory: divisibility tests, congruences, and proofs.
Negative numbers and modulo
Different systems define remainder slightly differently for negatives. In this calculator, we use the Euclidean remainder, which is always between 0 and n−1 (for n > 0).
Example: -13 mod 5 = 2, because -13 = (-3)×5 + 2.
Quick examples
- 37 mod 10 = 7
- 100 mod 9 = 1
- 256 mod 16 = 0
- -21 mod 4 = 3
Congruence basics
Two integers a and b are congruent modulo n if they leave the same remainder when divided by n. We write this as:
a ≡ b (mod n)
Equivalent statement: n divides (a − b). This idea is fundamental in algebra and cryptography.
FAQ
Can I use decimals?
No. This mod rechner is built for integers.
Can modulus be zero?
No. Modulo by zero is undefined.
Will very large numbers work?
Yes. The calculator uses BigInt internally for precise integer arithmetic.
Final note
If you are learning coding, math, or cryptography, mastering modulo is a huge win. Keep this page bookmarked as your quick, reliable modulo helper.