What the Modulo Calculator does
This modulo calculator gives the remainder of a divided by n - a mod n - together with the quotient, exactly, for numbers of any size up to 2,000 digits and for decimals such as 5.5. It shows all three conventions side by side, because -17 mod 5 is 3 in mathematics, Python and Excel but -2 in C, Java and JavaScript, and it also does modular exponentiation (a^e mod n) and the modular inverse with the working shown.
All arithmetic uses exact big integers in your browser, so there is no floating-point rounding: 0.3 mod 0.1 is 0, not 0.09999. Nothing you type is sent anywhere.
How to use it
- Enter the number a and the divisor n. Negative numbers, decimals and very long numbers are all fine; spaces or commas between digits are ignored.
- Choose how you want the headline answer: the mathematical remainder (never negative), the floored remainder used by Python and Excel, or the truncated remainder of C, Java and JavaScript's %. All three are shown underneath either way.
- Read the equation a = q x n + r to see the quotient and remainder fit together, and the language table to see what your programming language would return.
- For modular exponentiation, type an exponent: the tool computes a^e mod n by square-and-multiply and lists the steps when the exponent is small enough.
- Leave "Also find the inverse" ticked to get a^-1 mod n with the extended Euclidean algorithm, or a clear statement that none exists because gcd(a, n) is not 1.
Reading the results
Every convention satisfies a = q x n + r with |r| smaller than |n|. They differ only in which way the quotient is rounded: towards zero (truncated), downwards (floored) or so that r is never negative (Euclidean). For positive a and n they are identical.
If you are writing code, pick the row for your language. Bugs from this difference are common: using % to wrap an array index with a negative offset in JavaScript or Java gives a negative index, where Python would wrap correctly.
In modular arithmetic proper - clock arithmetic, residues, cryptography - the Euclidean remainder is the one meant: -17 is congruent to 3 modulo 5.
A modular inverse x satisfies a x x = 1 (mod n). It exists only when a and n share no common factor; the working table shows each step of the extended Euclidean algorithm, where every row keeps r = s x a + t x n.
Worked example: -17 mod 5, and a small RSA calculation
17 divided by 5 is 3 remainder 2, because 3 x 5 + 2 = 17. For -17 the conventions split. Truncated division rounds -3.4 towards zero to -3, so r = -17 - (-3 x 5) = -2. Floored division rounds down to -4, so r = -17 - (-4 x 5) = 3. The Euclidean remainder must be non-negative, which also gives 3.
So -17 % 5 is -2 in JavaScript, Java and C, and -17 % 5 is 3 in Python, as is =MOD(-17,5) in Excel. With a negative divisor, 17 mod -5 is 2 truncated but -3 floored, because the floored remainder takes the sign of the divisor.
For RSA with p = 61 and q = 53, n = 3233 and the public exponent is 17. Encrypting the message 65 is 65^17 mod 3233 = 2790. The private exponent is the inverse of 17 modulo (61 - 1) x (53 - 1) = 3120, which the extended Euclidean algorithm finds as 2753 because 17 x 2753 = 46,801 = 15 x 3120 + 1.
Decrypting, 2790^2753 mod 3233 = 65 again. The calculator never writes out 2790^2753 - a number with over 9,000 digits - because square-and-multiply reduces modulo 3233 after every step.
Formulas and scoring rules
- Division identity
a = q x n + r, |r| < |n|- Truncated
q = trunc(a / n); r = a - n x q (r has the sign of a)- Floored
q = floor(a / n); r = a - n x q (r has the sign of n)- Euclidean
r = a - |n| x floor(a / |n|); 0 <= r < |n|- Modular exponentiation
a^e mod n by square-and-multiply: for each bit of e, result = result x a^(2^i) mod n when the bit is 1A negative e is computed as (a^-1)^|e| mod n.- Modular inverse
extended Euclid: r_(i+1) = r_(i-1) - q_i r_i, likewise s and t; inverse = s mod n when gcd = 1Decimals are handled by scaling both numbers to whole numbers first, so results are exact; nothing is rounded.
Limitations: what the result does not prove
- Numbers are limited to 2,000 digits each so the page stays responsive. That is far beyond everyday needs and enough for textbook cryptography, but real cryptographic code should use a vetted library, not a web page.
- Modular exponentiation and the inverse need whole numbers; decimals are accepted only for the plain remainder.
- The language table describes the built-in operators and functions as they are commonly documented. Libraries and newer language versions may add other functions (for example Euclidean variants), so check your language's documentation for edge cases.
- Scientific notation such as 1e6 is not accepted, to avoid ambiguity; type the digits.
Privacy: where your data goes
Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.
Standards and sources
- ECMAScript Language Specification - the remainder operator
- Python documentation - binary arithmetic operations (%)
- Microsoft Support - MOD function
- Daan Leijen - Division and Modulus for Computer Scientists
Frequently asked questions
How do you calculate a mod n?
Divide a by n, keep the whole-number part of the quotient, and subtract that many n's from a. For 47 mod 6: 47 / 6 = 7.83, so q = 7, and 47 - 7 x 6 = 5. The remainder 5 is the answer, and 47 = 7 x 6 + 5 checks it.
What is the mod of a negative number?
It depends on the convention. In mathematics, Python and Excel, -17 mod 5 is 3, because -17 = -4 x 5 + 3. In C, C++, Java and JavaScript, -17 % 5 is -2, because those languages round the quotient towards zero. Both satisfy a = q x n + r; this page shows both side by side.
Why does JavaScript give a negative remainder?
Because % in JavaScript is a remainder operator that truncates the quotient towards zero, so the result takes the sign of the left-hand number. To get the mathematical result for a positive n, use ((a % n) + n) % n, or BigInt arithmetic for large numbers.
How do I calculate modulo of very large numbers?
Use exact integer arithmetic rather than a normal calculator, which rounds after about 15 to 17 significant digits. Type or paste the full number here - up to 2,000 digits - and the remainder is computed exactly with big integers in your browser.
How do I calculate modular exponentiation like 7^560 mod 561?
Use square-and-multiply: write the exponent in binary, repeatedly square the base modulo n, and multiply in the squares where the bit is 1, reducing modulo n each time. Enter 7 as a, 561 as n and 560 as the exponent; the answer is 1, which is why 561 is a famous Carmichael number.
How do I find the modular inverse?
Run the extended Euclidean algorithm on a and n. If the greatest common divisor is 1, the coefficient of a in the final row is the inverse, adjusted to lie between 0 and n - 1. For example, 3 x 4 = 12 = 1 (mod 11), so the inverse of 3 mod 11 is 4. If the gcd is not 1, no inverse exists.
Can you take the modulo of a decimal number?
Yes. The same identity a = q x n + r applies with a whole-number quotient: 5.5 mod 2 = 1.5, since 5.5 = 2 x 2 + 1.5. This calculator does it exactly, so 0.3 mod 0.1 gives 0, where floating-point arithmetic in many languages gives 0.09999999999999998.
Last reviewed by the A2Z.Tools team against the sources listed above.
