Calculate modulo operation (a mod b) with step-by-step solutions. Find remainders, understand division, and verify results with detailed mathematical explanations.
Modulo Operation:
a mod b = remainder
Returns the remainder after division
Enter values and click Calculate to see results
Find the remainder when dividing two numbers
The modulo operation (often abbreviated as "mod") is a fundamental mathematical operation that finds the remainder after division of one number by another. Our comprehensive modulo calculator helps you quickly compute a mod b with detailed step-by-step solutions, making it perfect for students, programmers, mathematicians, and anyone working with modular arithmetic or number theory.
When we write "a mod b", we're asking: "What is the remainder when a is divided by b?" For example, 5 mod 2 = 1 because when you divide 5 by 2, you get 2 with a remainder of 1. The modulo operation is extensively used in computer science, cryptography, clock arithmetic, and many areas of mathematics.
This calculator provides not just the answer, but also shows you the complete division process, verifies the result using the division algorithm, and explains each step clearly. Whether you're learning modular arithmetic for the first time, debugging code, or working on number theory problems, this tool provides accurate results with full transparency.
The modulo operation returns the remainder of a division operation. When you divide a number (dividend) by another number (divisor or modulus), you get a quotient and a remainder. The modulo operation specifically gives you that remainder.
For integers a and b (where b ≠ 0):
a mod b = r
Where r is the remainder such that:
a = q × b + r
where q is the quotient and 0 ≤ r < |b|
For example, 17 mod 5 = 2 because 17 = 3 × 5 + 2. The quotient is 3, and the remainder (modulo result) is 2. The remainder is always less than the divisor and is typically non-negative in most mathematical and programming contexts.
Input the number you want to divide. This is the number you're finding the remainder for.
Input the divisor (must be non-zero). This is the number you're dividing by.
The calculator will compute the modulo result and display the quotient, remainder, and verification.
Examine the answer, proof section, step-by-step solution, and division breakdown to understand the complete process.
Check the verification equation to confirm that Quotient × Divisor + Remainder = Dividend.
Result: 17 mod 5 = 2
Result: -17 mod 5 = 3
Used extensively in programming for array indexing, hash functions, circular buffers, and range wrapping. For example, to cycle through array indices or create periodic patterns.
Fundamental to encryption algorithms like RSA and Diffie-Hellman. Modular arithmetic provides the mathematical foundation for secure communication and digital signatures.
Time calculations use modulo 12 or 24. For example, 15:00 + 10 hours = 1:00 (next day) uses modulo to wrap around the clock face.
Determining day of the week, leap years, and date arithmetic. Modulo 7 is used to find which day of the week a particular date falls on.
Credit cards, ISBNs, and barcodes use modulo operations for error detection. The check digit is calculated using modulo to verify data integrity.
Musical intervals and chord progressions use modulo 12 (chromatic scale). Used to calculate note relationships and transpositions in music composition.
Wrong: Thinking 17 mod 5 = 3.4 (the division result).
Correct: Modulo gives only the remainder, not the quotient. 17 mod 5 = 2.
Wrong: Assuming -17 mod 5 = -2.
Correct: In most systems, modulo with positive divisor gives non-negative result: -17 mod 5 = 3.
Wrong: Attempting to calculate a mod 0.
Correct: Modulo by zero is undefined, just like division by zero. Always ensure divisor ≠ 0.
Wrong: Expecting remainder ≥ divisor.
Correct: The remainder is always less than the absolute value of the divisor: 0 ≤ r < |b|.
Wrong: Assuming all programming languages handle modulo identically.
Correct: Different languages handle negative numbers differently. Python vs C/Java have different modulo behaviors.
Wrong: Trying 17.5 mod 5.2 and expecting integer results.
Correct: Modulo is primarily defined for integers. For floats, use remainder functions or convert to integers first.
For small numbers, count up by the divisor until you exceed the dividend, then find the difference. Example: 17 mod 5 → 5, 10, 15, 20... 17 is between 15 and 20, so 17 - 15 = 2.
Remember: (a + b) mod n = ((a mod n) + (b mod n)) mod n. This property is useful for simplifying large calculations in cryptography and number theory.
Check if a number is even or odd using mod 2. If n mod 2 = 0, it's even; if n mod 2 = 1, it's odd. Simple and efficient!
Use modulo for wraparound indexing in programming: nextIndex = (currentIndex + 1) % arrayLength. This creates circular buffer behavior automatically.
Convert values to a specific range [0, n-1] using modulo. Useful for normalizing angles, creating periodic patterns, and constraining values.
If a mod b = 0, then a is divisible by b with no remainder. Quick way to test if one number divides evenly into another.
a % b - Always returns result with same sign as divisor
17 % 5 = 2, -17 % 5 = 3, 17 % -5 = -3
a % b - Returns result with same sign as dividend
17 % 5 = 2, -17 % 5 = -2, 17 % -5 = 2
a % b - Returns result with same sign as dividend
17 % 5 = 2, -17 % 5 = -2, 17 % -5 = 2
⚠️ Important Note:
Different programming languages implement modulo differently, especially for negative numbers. Always check your language's documentation to understand its specific behavior.
Test your understanding with these practice problems. Use our calculator to check your answers!
Problem 1: Beginner
Calculate 23 mod 7Hint: How many times does 7 go into 23?
Problem 2: Beginner
Calculate 100 mod 10Hint: Is 100 evenly divisible by 10?
Problem 3: Intermediate
What time will it be 100 hours from now if it's currently 3:00? Use modulo.Hint: Use mod 24 for 24-hour clock.
Problem 4: Intermediate
Calculate -15 mod 4Hint: Result should be non-negative.
Problem 5: Advanced
If today is Wednesday, what day will it be 100 days from now?Hint: Use mod 7 for days of the week.
Problem 6: Challenge
Find the last digit of 7^100 (7 to the power 100)Hint: Look for a pattern in powers of 7 modulo 10.
In mathematics, modulo and remainder are often used interchangeably, but there can be subtle differences in how negative numbers are handled. The modulo operation typically returns a non-negative result between 0 and |b|-1, while remainder may match the sign of the dividend. For positive numbers, they are identical. In this calculator, we follow the mathematical convention where modulo results are non-negative.
Modulo is extensively used in programming for: cycling through arrays (index % length), checking even/odd numbers (n % 2), creating hash functions, implementing circular buffers, generating periodic patterns, and constraining values to specific ranges. It's fundamental to many algorithms and data structures. Most programming languages use the % operator for modulo.
Modular arithmetic is the foundation of modern cryptography. RSA encryption, Diffie-Hellman key exchange, and elliptic curve cryptography all rely heavily on modulo operations. The difficulty of certain mathematical problems in modular arithmetic (like discrete logarithm) provides the security for these systems. Modulo operations ensure calculations stay within finite fields, making them suitable for computer implementation.
In mathematics, the modulo operation conventionally returns a non-negative result. However, different programming languages handle this differently. Python always returns a non-negative result matching the divisor's sign. Languages like C, Java, and JavaScript return a result matching the dividend's sign, which can be negative. This calculator follows the mathematical convention of non-negative results.
Modulo by zero is undefined, just like division by zero. There's no mathematical meaning to finding the remainder when dividing by zero. In programming, attempting modulo by zero typically causes a runtime error or exception. Always ensure your divisor (modulus) is non-zero before performing the operation. This calculator will alert you if you try to calculate modulo 0.
To calculate a mod b by hand: (1) Divide a by b to get quotient q and remainder r, (2) The remainder r is your answer. For example, 17 mod 5: divide 17 by 5 to get 3 remainder 2, so 17 mod 5 = 2. You can verify: 3 × 5 + 2 = 17. For negative numbers, adjust to ensure the result is non-negative and less than the absolute value of the divisor.
For a mod b where b is positive, the result always falls in the range [0, b-1]. For example, any number mod 5 will be 0, 1, 2, 3, or 4. This is why modulo is useful for creating cyclic patterns and constraining values to specific ranges. The result is always strictly less than the divisor's absolute value.
Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" after reaching a certain value (the modulus). It's like clock arithmetic—after 12 comes 1, not 13. In modular arithmetic, two numbers are considered equivalent (congruent) if they have the same remainder when divided by the modulus. Written as a ≡ b (mod n), meaning a and b differ by a multiple of n.
While modulo is primarily defined for integers, some programming languages support modulo with floating-point numbers (called fmod or remainder). However, results may be imprecise due to floating-point rounding errors. For mathematical clarity, it's best to use modulo with integers. If you need to work with decimals, consider converting to integers (by multiplying), performing modulo, then converting back.
For large numbers, use modular arithmetic properties: (a × b) mod n = ((a mod n) × (b mod n)) mod n, and (a + b) mod n = ((a mod n) + (b mod n)) mod n. These allow you to reduce numbers at each step, preventing overflow. For powers, use modular exponentiation (square-and-multiply algorithm). These techniques are crucial in cryptography and competitive programming.
Our Modulo Calculator provides a comprehensive, user-friendly tool for computing modulo operations with complete transparency and educational value. Whether you're a mathematics student learning number theory, a programmer working with algorithms, a cryptography enthusiast, or anyone needing to calculate remainders, this calculator delivers accurate results with detailed explanations.
Understanding the modulo operation is fundamental to many areas of mathematics and computer science. By mastering modulo calculations, you develop essential skills for programming, cryptography, discrete mathematics, and problem-solving. The step-by-step solutions and verification provided by our calculator help you learn the methodology and understand the underlying principles.
Start using our calculator today to compute modulo operations quickly, verify your work, debug your code, or explore the fascinating world of modular arithmetic. With proper mathematical notation, clear proofs, division breakdowns, and comprehensive explanations, mastering the modulo operation has never been easier. Make this calculator your essential tool for all modulo calculations and build confidence in your mathematical and programming abilities!