Modulo Calculator

Frequently Asked Questions

What does modulo mean?

a mod n is the remainder after dividing a by n. 17 mod 5 = 2. It answers "what is left over" and is central to clock arithmetic, hashing, and parity.

How are negative numbers handled?

Conventions differ: truncated (follows the sign of a, like JavaScript %), floored (follows the sign of n), and Euclidean (always non-negative). The tool shows the result for the selected convention.

What is modulo used for?

Even/odd tests (x mod 2), cycling through arrays, days-of-week math, check digits, cryptography, and hashing. Any "wrap-around" problem is modulo.

Is mod the same as the remainder operator?

For positive operands, yes. For negatives they can differ - many languages' % is the truncated remainder, while "mod" in math is usually the floored or Euclidean result.

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How This Calculator Works

The modulo operation returns the remainder of a ÷ n. Every convention starts from the division identity a = n × q + r; what differs is how the quotient q is rounded, which fixes the sign of r. Truncated (used by C and JavaScript's %) rounds q toward zero, so r takes the sign of the dividend a. Floored (used by Python and most mathematics) rounds q toward −∞, so r takes the sign of the divisor n. Euclidean always yields 0 ≤ r < |n| regardless of signs, which is the convention number theory and modular arithmetic assume. For positive a and n all three agree; they only diverge when a or n is negative. This tool computes all three and highlights the one you select so you can match whatever language or textbook you are working from.

Worked Example: −17 mod 5

Take a = −17, n = 5. Truncated: q = trunc(−17 ÷ 5) = trunc(−3.4) = −3, so r = −17 − 5×(−3) = −17 + 15 = −2. Floored: q = floor(−3.4) = −4, so r = −17 − 5×(−4) = −17 + 20 = 3. Euclidean: r must be in [0, 5), giving r = 3 and q = −4 here (it coincides with floored when n > 0). So the same expression yields −2, 3, or 3 depending on convention - a frequent source of bugs when porting code between C and Python. Verify the Euclidean result: 5 × (−4) + 3 = −20 + 3 = −17. ✓

Where Modulo Is Used

Clock & calendar math: hours wrap mod 12 or 24, weekdays mod 7 - "what day is 100 days from Tuesday?" is (2 + 100) mod 7.

Hashing & data structures: hash tables map a key to a bucket with key mod tableSize; ring buffers index mod capacity.

Cryptography: RSA, Diffie-Hellman, and elliptic-curve schemes are built entirely on modular exponentiation in ℤ_n.

Parity & cycles: n mod 2 tests even/odd; check digits (ISBN, IBAN, Luhn) and zebra-striping use modulo.

Common Mistakes & Edge Cases

Assuming % is always non-negative. In C, Java, and JavaScript, −17 % 5 is −2, which breaks array indexing - use ((a % n) + n) % n for a safe non-negative index.

Confusing the floored and Euclidean results when n is negative: they agree only when n > 0.

n = 0 is undefined - this tool rejects it rather than returning NaN.

Using modulo on floats expecting integer behavior; here inputs are coerced numerically but the identity still holds for the chosen rounding.

Forgetting that a ≡ b (mod n) means n divides (a − b), not that they share the same printed remainder under a language's native operator.

Related Concepts & Limits

Modulo is the gateway to modular arithmetic: addition, multiplication, and exponentiation all stay well-defined mod n, enabling fast modular exponentiation and the Chinese Remainder Theorem. The greatest-common-divisor Euclidean algorithm is just repeated modulo: gcd(a, b) = gcd(b, a mod b). Modular inverses (the basis of RSA decryption) exist only when gcd(a, n) = 1. This calculator works on JavaScript doubles, so magnitudes beyond about 9 × 10¹⁵ lose exactness - for big-integer modular work use the Big Number Calculator. The three conventions shown here cover essentially every programming language and textbook you are likely to encounter.

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