Numbers
Calculadora de aritmética modular
Used by developers, writers, and creators worldwide.
A modular arithmetic calculator computes a number modulo a chosen modulus, giving the remainder that wraps around like the hours on a clock. Enter a value and a modulus and it returns the least non-negative remainder, the floor quotient with the division written out, the raw result of the percent operator, and the congruence statement. Modular arithmetic underpins clocks, calendars, hashing, checksums, cryptography, and any cyclic pattern. A subtle trap is that many programming languages return a negative remainder for negative inputs, so the tool also gives the proper least non-negative remainder mathematicians expect, computed safely. Students use it to learn congruences, programmers to reason about wrap-around behaviour, and puzzle solvers to handle cyclic problems. Seeing both the mathematical remainder and the raw operator output side by side makes the common negative-number pitfall obvious. Use it to compute a clock-style remainder or to check how a value reduces modulo m.
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How to use
- Choose your options above
- Click Generate
- Copy your result
Detailed instructions
- Enter the number you want to reduce.
- Enter the modulus to divide by.
- Click Generate to compute the remainder.
- Compare the mathematical and raw operator results.
Use Cases
- •Computing a remainder for clock or calendar problems
- •Learning congruences and modular arithmetic
- •Checking wrap-around behaviour in code
- •Reasoning about checksums and hashing
- •Handling negative numbers under a modulus correctly
Tips
- →The least non-negative remainder is always between 0 and m minus 1.
- →Watch for negative results from a raw percent operator.
- →Think of the modulus as the size of a clock face.
- →Congruent numbers behave identically under that modulus.
FAQ
why are there two remainder values
Mathematics defines the modulo result as a least non-negative remainder, but many languages’ percent operator keeps the sign of the input. The tool shows both so you can see the difference, which matters most with negative numbers.
how is the negative case handled
For a negative input the tool computes the remainder, adds the modulus, and takes the modulus again, guaranteeing a result between zero and one less than the modulus, which is the conventional mathematical answer.
what does congruent mean here
Two numbers are congruent modulo m when they leave the same remainder on division by m. The congruence line states that your number and its remainder are interchangeable in modular arithmetic with that modulus.
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