Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Basic Math Operations > Modulo Calculator

Modulo Calculator

Calculate modulo (remainder) with step-by-step division process, interactive visual diagrams, and support for integers, decimals, negative numbers, and scientific notation.

Modulo Calculator

Try these examples:

17 mod 5 100 mod 7 -17 mod 5 8.5 mod 2.5 10^6 mod 13 25 mod 25
mod

Embed Modulo Calculator Widget
Please make sure the form is complete and valid.

About Modulo Calculator

Welcome to the Modulo Calculator, a comprehensive free online tool for calculating the modulo (remainder) of any two numbers. This calculator provides step-by-step division breakdowns, interactive visual diagrams, and supports integers, decimals, negative numbers, and scientific notation. Whether you are learning mathematics, programming, or solving cryptography problems, this tool makes modulo operations clear and easy to understand.

What is Modulo (Mod) Operation?

The modulo operation (often written as mod or %) finds the remainder after dividing one number (the dividend) by another (the divisor). It answers the question: "After dividing a by n, what is left over?"

Modulo Definition
$a \mod n = r$ where $a = n \times q + r$ and $0 \le r < |n|$

Here, $a$ is the dividend, $n$ is the divisor, $q$ is the quotient (integer part of division), and $r$ is the remainder (the modulo result).

Example: 17 mod 5

17 divided by 5 = 3 with remainder 2

Because: 17 = 5 × 3 + 2

Therefore: 17 mod 5 = 2

How to Calculate Modulo

  1. Enter the dividend (a): Input the number you want to divide. This can be positive, negative, a decimal, or in scientific notation (e.g., 1.5e10).
  2. Enter the divisor (n): Input the number you are dividing by. This cannot be zero, but can be positive, negative, or a decimal.
  3. Click Calculate Modulo: Press the button to see your result with a complete step-by-step breakdown.
  4. Review the results: See the remainder, quotient, verification equation, and (for simple positive integers) a visual diagram showing the grouping.

Manual Calculation Steps

To calculate $a \mod n$ manually:

  1. Divide: Calculate $a \div n$
  2. Floor: Take the floor (round toward negative infinity) to get quotient $q = \lfloor a/n \rfloor$
  3. Multiply: Calculate $n \times q$
  4. Subtract: Calculate remainder $r = a - n \times q$
Example: Calculate 23 mod 7

Step 1: 23 ÷ 7 = 3.2857...

Step 2: q = floor(3.2857) = 3

Step 3: 7 × 3 = 21

Step 4: r = 23 - 21 = 2

Common Uses of Modulo

🔢
Even/Odd Check
n mod 2 = 0 means n is even; n mod 2 = 1 means n is odd. This is the most common modulo use in programming.
🕐
Clock Arithmetic
Convert 24-hour to 12-hour format: 14 mod 12 = 2 (2:00 PM). Calculate time wrapping around midnight.
🔄
Cyclic Patterns
Create repeating sequences, circular arrays, and round-robin scheduling. Index i mod n ensures staying within bounds.
🔐
Cryptography
RSA encryption, Diffie-Hellman key exchange, and hash functions all rely heavily on modular arithmetic.
📊
Hash Functions
hash(key) mod table_size determines where to store data in hash tables, ensuring indices stay within array bounds.
📅
Calendar Calculations
Determine day of week, leap years, and date arithmetic. Days repeat every 7, so day mod 7 gives the weekday.

Modulo with Different Number Types

Positive Integers

For positive integers, modulo is straightforward: the remainder is always between 0 and n-1.

Negative Numbers

Negative numbers can be tricky because different systems define modulo differently. This calculator uses the mathematical definition where the remainder is always non-negative (0 to |n|-1):

Programming vs Math Convention

Programming languages vary in handling negative modulo:

Python: -17 % 5 = 3 (floored division - matches math)

JavaScript/C/Java: -17 % 5 = -2 (truncated division)

Decimal Numbers

Modulo extends to decimal (floating-point) numbers using the same principle:

Scientific Notation

This calculator supports scientific notation for very large or small numbers:

Modulo Properties and Rules

Fundamental Properties

Arithmetic with Modulo

Modular Arithmetic Rules

$(a + b) \mod n = ((a \mod n) + (b \mod n)) \mod n$

$(a - b) \mod n = ((a \mod n) - (b \mod n) + n) \mod n$

$(a \times b) \mod n = ((a \mod n) \times (b \mod n)) \mod n$

These properties are essential in cryptography and computer science, allowing calculations with very large numbers without overflow.

Modulo vs Division vs Remainder

Division (÷ or /)

Division gives the quotient, which can be a decimal: 17 ÷ 5 = 3.4

Integer Division (// or div)

Integer division gives only the whole number part: 17 // 5 = 3

Modulo (mod or %)

Modulo gives only the remainder: 17 mod 5 = 2

Relationship

The Division Identity
$a = n \times (a \div n) + (a \mod n)$

For 17 and 5: 17 = 5 × 3 + 2 ✓

Frequently Asked Questions

What is modulo (mod) operation?

The modulo operation (often abbreviated as mod) finds the remainder after division of one number by another. For example, 17 mod 5 = 2 because 17 divided by 5 equals 3 with a remainder of 2. Mathematically: a mod n = r where a = n × q + r and 0 ≤ r < |n|.

How do you calculate modulo?

To calculate a mod n: 1) Divide a by n and find the integer quotient q = floor(a/n). 2) Multiply q by n. 3) Subtract from a to get the remainder: r = a - n × q. For example, 17 mod 5: q = floor(17/5) = 3, r = 17 - 5 × 3 = 17 - 15 = 2.

What is the difference between mod and remainder?

For positive numbers, modulo and remainder are identical. The difference appears with negative numbers. In mathematics, modulo always returns a non-negative result (0 ≤ r < |n|), while the remainder can be negative depending on the programming language. This calculator uses the mathematical definition.

What are common uses of modulo operation?

Modulo is used in: 1) Checking if a number is even/odd (n mod 2), 2) Clock arithmetic (24-hour to 12-hour conversion), 3) Cyclic patterns and circular arrays, 4) Hash functions and cryptography, 5) Generating pseudo-random numbers, 6) Determining divisibility, 7) Calendar calculations.

How does modulo work with negative numbers?

With negative numbers, different conventions exist. In mathematics and this calculator, the result is always non-negative: -17 mod 5 = 3 (not -2). This is because -17 = 5 × (-4) + 3. Some programming languages return -2 using truncated division. Understanding this difference is crucial for programming.

Can modulo work with decimal numbers?

Yes, modulo can be extended to decimal (floating-point) numbers. For example, 7.5 mod 2.5 = 0 because 7.5 = 2.5 × 3 + 0. And 8.7 mod 2.5 = 1.2 because 8.7 = 2.5 × 3 + 1.2. This calculator supports decimal modulo calculations with high precision.

Additional Resources

Reference this content, page, or tool as:

"Modulo Calculator" at https://MiniWebtool.com/modulo-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 05, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Quotient and Remainder CalculatorFeatured

Basic Math Operations:

Top & Updated:

Random PickerRandom Name PickerRelative Standard Deviation CalculatorFPS ConverterSort NumbersMAC Address GeneratorRemove SpacesFeet and Inches to Cm ConverterMAC Address LookupBitwise CalculatorWord to Phone Number ConverterLine CounterBatting Average CalculatorPercent Off CalculatorSum CalculatorFacebook User ID LookupRandom Quote GeneratorLog Base 10 CalculatorNumber of Digits CalculatorSHA256 Hash GeneratorSun, Moon & Rising Sign Calculator 🌞🌙✨Saturn Return CalculatorRoman Numerals ConverterDecimal to BCD ConverterRandom Truth or Dare GeneratorRandom Credit Card GeneratorERA CalculatorBinary to Gray Code ConverterAI ParaphraserVideo to Image ExtractorOne Rep Max (1RM) CalculatorCm to Feet and Inches ConverterQuotient and Remainder CalculatorSalary Conversion CalculatorMP3 LooperPhone Number ExtractorOutlier CalculatorBCD to Decimal ConverterOctal CalculatorCompound Growth CalculatorRemove Audio from VideoOn Base Percentage CalculatorFirst n Digits of PiSquare Root (√) CalculatorNumber to Word ConverterGray Code to Binary ConverterDay of Year CalendarTime Duration CalculatorAI Punctuation AdderMerge VideosRemove AccentCompare Two StringsScientific Notation to Decimal ConverterBinary to BCD ConverterCM to Inches ConverterModulo CalculatorLeap Years ListText FormatterIP Address to Hex ConverterRandom Group GeneratorConvolution CalculatorAntilog CalculatorSlugging Percentage CalculatorMaster Number CalculatorMedian CalculatorList of Prime NumbersStandard Error CalculatorPercentage Increase CalculatorAudio SplitterPER CalculatorPercent Growth Rate CalculatorAverage Deviation CalculatorMaze GeneratorImage SplitterAdd or Replace Audio in VideoRandom Birthday GeneratorBreak Line by CharactersRandom Tournament Bracket GeneratorOPS CalculatorLove Compatibility Calculator📅 Date CalculatorBonus CalculatorRandom Emoji GeneratorRandomize LinesWhat is my Lucky Number?Video SplitterPVIF CalculatorNatural Log CalculatorProportion CalculatorSum of Positive Integers CalculatorMAC Address AnalyzerRemove Leading Trailing SpacesBCD to Binary ConverterSHA512 Hash GeneratorFirst n Digits of eURL ExtractorVideo CropperNumber ExtractorPVIFA CalculatorLog Base 2 CalculatorEmail ExtractorAdd Prefix and Suffix to TextWHIP CalculatorDay of the Year Calculator - What Day of the Year Is It Today?Video CompressorSort Lines AlphabeticallyHex to BCD ConverterLottery Number GeneratorBCD to Hex ConverterList RandomizerAverage CalculatorReverse VideoWAR CalculatorHypotenuse CalculatorActual Cash Value CalculatorAngel Number CalculatorRoot Mean Square CalculatorSHA3-256 Hash GeneratorAI Sentence ExpanderLbs to Kg ConverterHex to Decimal ConverterRandom String GeneratorAmortization CalculatorMarkup CalculatorName Number CalculatorDecimal to Hex ConverterList of Fibonacci NumbersRandom Number PickerBlood Donation Time CalculatorField Goal Percentage CalculatorRandom Line PickerRatio to Percentage CalculatorAPY to APR CalculatorSort Text By LengthRandom Object GeneratorName RandomizerRandom Animal GeneratorSquare Numbers ListFixed Deposit CalculatorPPM to Percent ConverterCube Numbers ListCaesar Cipher ToolMorse Code GeneratorPercent to PPM ConverterSRT to TXT Converterkg to lbs ConverterXML ValidatorArithmetic Mean CalculatorBoard Foot CalculatorSquareness Checker (3-4-5 Rule) 📐Miter Angle Calculator 📐Slope and Grade Calculator 📐Sine CalculatorCosine CalculatorHigh-Precision Tangent CalculatorArcsin CalculatorArccos (Inverse Cosine) CalculatorArctan CalculatorArctan2 CalculatorCosecant/Secant/Cotangent CalculatorHyperbolic Functions CalculatorRight Triangle CalculatorGeneral Triangle SolverLaw of Sines CalculatorLaw of Cosines CalculatorDMS to Decimal Degrees ConverterDecimal Degrees to DMS ConverterPolar to Cartesian ConverterHigh-Precision Cartesian to Polar Coordinates ConverterInteractive Unit Circle VisualizerTrigonometric Identities CalculatorTrigonometric Function GrapherAlgebraic Expression SimplifierRadical SimplifierRational Expression CalculatorPolynomial Factoring CalculatorExpand Polynomials CalculatorPolynomial Long Division CalculatorSynthetic Division CalculatorLinear Equation SolverAbsolute Value Equation SolverRadical Equation SolverSystem of Linear Equations SolverInequality SolverAbsolute Value Inequality SolverSystem of Inequalities GrapherFunction GrapherDomain and Range CalculatorInverse Function CalculatorFunction Composition CalculatorVertex and Axis of Symmetry CalculatorX and Y Intercept CalculatorHTML Entity Encoder/DecoderURL Slug GeneratorJSON String Escape/UnescapeMarkdown Table GeneratorSmart Quotes RemoverDaily Time Savings CalculatorLorem Ipsum GeneratorReadability Score CalculatorKeyword Density CheckerBionic Reading ConverterText to Speech ReaderFancy Text GeneratorZalgo Text GeneratorUpside Down Text GeneratorInvisible Text GeneratorASCII Art GeneratorList Difference CalculatorText Column ExtractorText to SQL List ConverterInvisible Character RemoverStock Average CalculatorPosition Size CalculatorMargin Call CalculatorShort Selling Profit CalculatorRisk of Ruin CalculatorBlack-Scholes Option Pricing CalculatorOption Greeks CalculatorImplied Volatility CalculatorOptions Profit CalculatorImpermanent Loss CalculatorCrypto Arbitrage CalculatorSatoshi to USD ConverterCrypto Leverage CalculatorPivot Point CalculatorFibonacci Extension CalculatorStop Loss & Take Profit CalculatorKelly Criterion CalculatorMartingale Strategy CalculatorCompound Daily Interest CalculatorPip Value CalculatorGaussian Distribution GeneratorRandom Meal GeneratorRandom Color Palette GeneratorRandom Country GeneratorRandom JSON GeneratorRandom User-Agent GeneratorRandom Coordinate GeneratorRandom Date GeneratorRandom Time GeneratorRandom IP Address GeneratorRandom Playing Card GeneratorMagic 8-BallRock Paper Scissors GeneratorCoin FlipperDice RollerSpin the WheelAquarium Volume & Stocking CalculatorAquarium Substrate CalculatorScale Model Conversion CalculatorPlant Spacing CalculatorDice Roll Probability CalculatorDepth of Field (DoF) CalculatorGolden Hour / Blue Hour CalculatorPrint Size & Resolution Calculator (DPI/PPI)Soap Making Lye Calculator (SAP)Candle Wax & Fragrance Oil CalculatorHomebrew ABV Calculator (Alcohol by Volume)Coffee Brew Ratio CalculatorBrine and Salinity CalculatorYarn CalculatorFabric CalculatorCross-Stitch Size CalculatorBaker's Percentage CalculatorDough Hydration CalculatorBaking Pan Size ConverterYeast Conversion Calculator