Modulo Calculator

A modulo calculator returns what is left over after one number is divided by another. Written a mod n, it keeps only the leftover, not the quotient. Example: 17 mod 5 = 2, because 5 fits into 17 three whole times (15) and 2 remains. Negative inputs have two accepted answers, and this page shows both.

Three sister tools cover neighboring questions. The Long Division Calculator writes out the full division that produces the quotient and remainder step by step, the GCF Calculator runs the Euclidean algorithm, which is nothing but repeated modulo, and the LCM Calculator answers the mirror-image question of the smallest common multiple. Use this page when you want the leftover itself, especially when a minus sign is involved.

How the modulo operation works

Every division of a whole number a (the dividend) by a non-zero whole number n (the modulus or divisor) can be written with one quotient and one remainder:

Here q is the quotient and r is the remainder, the value the modulo operation returns. The formula alone is not quite enough, because it does not say how to round the quotient when the division does not come out even. That single choice is what separates the conventions:

• Floored: round the quotient down, toward negative infinity. The remainder then takes the sign of the modulus. This is what Python, Ruby and the MOD function in Excel and Google Sheets do.
• Truncated: round the quotient toward zero, simply dropping the fractional part. The remainder then takes the sign of the dividend. This is what C, C++, Java, C# and JavaScript do with the percent operator.
• Euclidean: choose the quotient so the remainder is never negative, whatever the sign of the modulus. This is the version mathematicians usually mean when they talk about residues.

When both numbers are positive, all three agree, which is why most people never notice the difference until a minus sign appears.

Worked example: −17 mod 5, step by step

Take the dividend a = −17 and the modulus n = 5. Work the floored version first:

1. Divide: −17 ÷ 5 = −3.4.
2. Round the quotient down to the next whole number: −4.
3. Multiply back: 5 × (−4) = −20.
4. Subtract: r = −17 − (−20) = 3.
5. Check the identity: 5 × (−4) + 3 = −20 + 3 = −17. Correct.

Now the truncated version, which is what most programming languages give you:

1. Divide: −17 ÷ 5 = −3.4.
2. Round the quotient toward zero, dropping the decimals: −3.
3. Multiply back: 5 × (−3) = −15.
4. Subtract: r = −17 − (−15) = −2.
5. Check the identity: 5 × (−3) + (−2) = −15 − 2 = −17. Also correct.

So −17 mod 5 is 3 in Python and in a spreadsheet, and −2 in JavaScript, Java or C. Neither answer is a bug. They differ by exactly one copy of the modulus, since 3 − 5 = −2, and both satisfy a = n × q + r.

Second worked example: 1,234 mod 12

With two positive numbers the arithmetic is simpler. Divide 1,234 by 12 to get 102.833..., drop the fraction for a quotient of 102, multiply back to 12 × 102 = 1,224, and subtract: 1,234 − 1,224 = 10. So 1,234 mod 12 = 10, and because the remainder is not zero, 1,234 is not a multiple of 12. Read on a 12-hour clock face, starting at 12 and counting forward 1,234 hours lands you 10 hours around the dial.

Negative numbers: floored, truncated and Euclidean side by side

The table below runs the dividend 17 and its negative against the modulus 5 and its negative, so you can see exactly where the three conventions split. Every row satisfies a = n × q + r in its own column pair.

Two patterns fall out of the table. The floored remainder always carries the sign of the modulus, and the truncated remainder always carries the sign of the dividend. When the two inputs share a sign, the rows agree.

1,234 mod 2 through mod 12

Running one number against every small modulus is a fast way to see its divisibility fingerprint. Here is 1,234 with the quotient and the floored remainder for each modulus from 2 to 12. The calculator above builds the same table for whatever number you type.

Only mod 2 gives zero, so 1,234 is even but shares no other factor from 2 to 12. Notice that mod 10 = 4 is simply the last digit, and mod 100 would be the last two digits, 34. Powers of ten make the modulo a digit-slicing tool.

Which convention does your tool use?

If two tools hand you different answers for the same negative input, this table usually explains why. It lists the sign rule each environment applies to the remainder.

If you need a non-negative result in a truncated language, the standard fix is to add the modulus and take the modulo again, writing the expression as a percent n plus n, then percent n once more. That maps −2 back to 3 without changing any positive result.

How to use this modulo calculator

Two numbers are all you need, and the result updates as you type. Work through the fields in order:

1. Dividend (a): the number being divided. Whole numbers of any length are handled exactly; decimals and negatives are accepted too.
2. Modulus (n): the number you divide by, sometimes called the divisor. It cannot be zero. Use the quick buttons for the common moduli 2, 7, 10, 12, 24, 60 and 100.
3. Sign convention: pick floored, truncated or Euclidean to choose which value appears as the headline. The other results stay visible in the tiles underneath, so you never have to run the calculation twice.
4. Read the check line: the identity a = n × q + r is printed out with your actual numbers, which is the fastest way to confirm a hand calculation.
5. Scan the table: the mod 2 through mod 12 table shows every small remainder at once, with a flag on any modulus that divides your number evenly.

Who this calculator is for

• Students checking homework on remainders, divisibility and modular arithmetic.
• Programmers debugging an index that wrapped the wrong way, or porting code between a floored language and a truncated one.
• Spreadsheet users who need to know why the MOD function disagrees with a result from their code.
• Puzzle and competition solvers working congruences, cycle lengths and last-digit questions.
• Anyone scheduling on a cycle, such as rotating shifts, recurring dates or a repeating rota, where the answer is a leftover position rather than a count.

Key terms explained

• Dividend (a): the number being divided, the value on the left of a mod n.
• Modulus (n): the number you divide by. It sets the size of the wrap, so a positive modulus n produces remainders from 0 to n minus 1.
• Quotient (q): how many whole copies of the modulus fit inside the dividend, after rounding in the chosen direction.
• Remainder (r): what is left over, and the value the modulo operation returns.
• Residue: another word for the remainder in modular arithmetic, usually the Euclidean one that is never negative.
• Congruence: the statement that two numbers leave the same remainder, as in 26 is congruent to 2 modulo 12.
• Divisor: a number that divides another with remainder 0. If a mod n = 0, then n is a divisor of a.

Where the modulo shows up in everyday life

Modulo is the arithmetic of anything that wraps around, which makes it far more common than its name suggests:

• Clocks: 26 mod 12 = 2, so 26 hours after noon the hour hand points at 2. Minutes and seconds run on mod 60, and 24-hour time on mod 24.
• Time conversion: 3,725 seconds is 3,725 mod 60 = 5 seconds left over, 62 whole minutes, and 62 mod 60 = 2 minutes left over after 1 hour, giving 1:02:05.
• Weekdays: 100 mod 7 = 2, so 100 days from a Monday is a Wednesday.
• Even and odd: a mod 2 = 0 means even, 1 means odd. That single test drives alternating table rows, striped charts and coin-flip logic.
• Leap years: the Gregorian rule is pure modulo. 2026 mod 4 = 2, so 2026 is not a leap year. 2100 mod 4 = 0 but 2100 mod 400 = 100, so 2100 is skipped, while 2000 mod 400 = 0 made 2000 a leap year.
• Splitting things up: 48 slices shared by 5 people leaves 48 mod 5 = 3 slices over.
• Programming: wrapping an index back to the start of an array, spreading keys across hash buckets, striping work across servers, and computing check digits on account, ISBN and card numbers.

Divisibility shortcuts you can do in your head

For small moduli you rarely need long division at all, because the remainder can be read off the digits:

• mod 2: look at the last digit. Even digit, remainder 0.
• mod 5 and mod 10: the last digit decides. 97 mod 10 = 7, and 97 mod 5 = 2.
• mod 4: only the last two digits matter, because 100 is a multiple of 4. For 1,234 that is 34, and 34 mod 4 = 2, which matches the table above.
• mod 3 and mod 9: add the digits. For 1,234 the digit sum is 10, and 10 mod 9 = 1, so 1,234 mod 9 = 1. The same digit sum gives 10 mod 3 = 1 for mod 3.
• Powers of 10: 123,456,789 mod 1,000 = 789, the last three digits.

What changes the result

If you are experimenting with the inputs, three things move the answer:

• The size of the modulus: it fixes the range of possible answers. A positive modulus n can only ever return 0 through n minus 1, so mod 7 answers live in 0 to 6 no matter how large the dividend is.
• The signs: the conventions only diverge when the dividend and the modulus have different signs. Two positives, or two negatives, and every method agrees.
• Adding or subtracting whole moduli: the remainder does not change. 23 mod 5, 28 mod 5 and 1,000,023 mod 5 all equal 3, because adding a multiple of the modulus adds nothing to the leftover. That is the whole idea of congruence.

Tips for getting the answer you actually want

• Decide the convention before you code. If a negative result would break your logic, use the Euclidean value and force it in code with the add-then-modulo trick.
• Verify with the identity. Multiply the quotient by the modulus and add the remainder. If you do not get the dividend back, the quotient was rounded the wrong way.
• Use the wrap deliberately. To cycle a counter through positions 0 to 5, increment it and take mod 6 every time rather than testing for the end value.
• Keep the modulus positive. A negative modulus is legal but the results are counter-intuitive, and almost every practical use has n greater than zero.
• Watch decimals. Binary floating point cannot store 0.1 exactly, so a decimal modulo can land a hair away from the value you expect. Scale to whole numbers when the exact answer matters.

Limitations and assumptions

• The modulus cannot be zero. There is no quotient that satisfies a = 0 × q + r for a non-zero a, so the operation is undefined.
• Whole-number inputs are exact at any length, but decimal inputs use standard floating-point arithmetic, precise to roughly 15 significant digits, and are rounded to ten decimal places for display.
• This page covers the single operation a mod n. It does not do modular exponentiation, modular inverses, or solving a congruence for an unknown.
• Results are pure arithmetic, so no rounding rules, tax tables or measurement standards apply. The only judgment call is the sign convention, and all three are shown.

How it compares to related calculators

This page answers "what is left over after dividing?" If your question is slightly different, another tool fits better:

• To see the whole division written out, digit by digit, use the Long Division Calculator.
• To find the largest number that divides two values with remainder 0, use the GCF Calculator, which applies repeated modulo through the Euclidean algorithm.
• To find the smallest number that two values both divide into, use the LCM Calculator.
• To break a number into its prime building blocks, use the Prime Factorization Calculator.
• To convert between number bases, where repeated modulo is exactly how digits are extracted, use the Binary Calculator.
• To turn a leftover into a fraction of the divisor, use the Fraction Calculator, or the Percentage Calculator to express it as a share.

Sources

Everything on this page is deterministic arithmetic and exact definition, so no external data source is needed. The definitions used are:

• The division algorithm, a = n × q + r with the remainder smaller in size than the modulus - standard number theory (Euclid's Elements, Book VII).
• U.S. Common Core State Standards for Mathematics (4.NBT.B.6 and 6.NS.B.2) - finding whole-number quotients and remainders.
• ECMA-262, the ECMAScript language specification - the percent operator returns a remainder carrying the sign of the dividend, which is the truncated convention used in this calculator.
• ISO/IEC 9899, the C language standard (C99 and later) - integer division truncates toward zero, so the percent operator follows the sign of the dividend.
• The Gregorian leap-year rule, divisible by 4 except centuries that are not divisible by 400, quoted in the everyday-uses section.