Modulo

Q: What does modulo (a mod b) mean?

Modulo, written as a mod b, gives the remainder when a is divided by b. For example, 17 mod 5 = 2 because 17 ÷ 5 = 3 with remainder 2. The relationship is always a = q x b + r, where q is the quotient (floor of a ÷ b) and r is the remainder (0 <= r < b for positive integers).

Q: What is the difference between modulo and remainder?

Modulo and remainder are closely related but not identical for negative numbers. Modulo always produces a non-negative result in mathematics (the remainder in Euclidean division), while the remainder operator (%) in many programming languages can return a negative value. This calculator follows the mathematical convention for positive integers where they produce the same result.

Q: How do you calculate a mod b step by step?

Step 1: Divide a by b to get the quotient. Step 2: Round the quotient down to the nearest integer (floor division). Step 3: Multiply the rounded quotient by b. Step 4: Subtract that product from a. The result is the remainder. For 250 mod 24: 250 ÷ 24 = 10.42 -> floor = 10; 10 x 24 = 240; 250 - 240 = 10. So 250 mod 24 = 10.

Find the modulo remainder of any division with a step-by-step breakdown. Free online modulo calculator for students, programmers, and math enthusiasts.

Compute a mod b

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About This Calculator

About the Modulo Calculator

The Modulo Calculator computes the remainder of a division between two positive integers, expressed as a mod b. Enter any dividend a and divisor b to get the remainder r where a = q x b + r and 0 <= r < b. This free online tool is designed for students learning modular arithmetic, programmers implementing cyclic logic, and anyone who needs to quickly find the remainder of a division.

The calculation follows the standard mathematical definition: divide a by b, round the quotient down to the nearest integer (floor division), multiply the quotient by b, and subtract from a. The result is displayed alongside the full equation, the quotient, and the remainder. Modulo operations are fundamental in number theory, computer science, and everyday applications such as telling time (mod 12) and validating credit card numbers (Luhn algorithm mod 10).

Regional Notes

India (IN): Modulo operations are taught in secondary mathematics under number theory and divisibility. The same mathematical definition applies globally.

United States (US): Modulo is introduced in pre-algebra and algebra courses. The % operator in most programming languages follows the remainder convention, which matches math modulo for positive integers as used here.

United Kingdom (UK): Modular arithmetic is part of the GCSE and A-Level mathematics curriculum. Common applications include clock arithmetic, cryptography, and computer science coursework.

Frequently Asked Questions

What does modulo (a mod b) mean?

Modulo, written as a mod b, gives the remainder when a is divided by b. For example, 17 mod 5 = 2 because 17 ÷ 5 = 3 with remainder 2. The relationship is always a = q x b + r, where q is the quotient (floor of a ÷ b) and r is the remainder (0 <= r < b for positive integers).

What is the difference between modulo and remainder?

Modulo and remainder are closely related but not identical for negative numbers. Modulo always produces a non-negative result in mathematics (the remainder in Euclidean division), while the remainder operator (%) in many programming languages can return a negative value. This calculator follows the mathematical convention for positive integers where they produce the same result.

How do you calculate a mod b step by step?

Step 1: Divide a by b to get the quotient. Step 2: Round the quotient down to the nearest integer (floor division). Step 3: Multiply the rounded quotient by b. Step 4: Subtract that product from a. The result is the remainder. For 250 mod 24: 250 ÷ 24 = 10.42 -> floor = 10; 10 x 24 = 240; 250 - 240 = 10. So 250 mod 24 = 10.

Is this modulo calculator free to use?

Yes, all calculators on Calculy are completely free to use. There are no hidden charges, account requirements, or usage limits.

What are common real-world uses of modulo?

Modulo is used in clock arithmetic (telling time), check digit validation (credit cards via Luhn algorithm, ISBN, IBAN), finding even/odd numbers (n mod 2), splitting items equally (pizza slices mod people), and in programming for circular buffers, hashing, and index wrapping. It is also fundamental in cryptography and number theory.

Can I compute modulo with a negative dividend?

This calculator accepts positive integers only. For computations with negative dividends or divisors, use our dedicated Modulo of Negative Numbers calculator, which explains the difference between the mathematical modulo and the programming remainder operator.

What is the formula for checking the modulo result?

To verify a modulo result, use the formula a = q x b + r, where a is the dividend, b is the divisor, q is the floor of a ÷ b, and r is the remainder. For example, for a = 17 and b = 5: q = 3, r = 2, and 17 = 3 x 5 + 2 checks out.

What is the difference between floor division and truncation in modulo?

Floor division rounds the quotient down to the nearest integer (toward negative infinity), while truncation removes the fractional part (toward zero). For positive numbers, they give the same result. This calculator uses floor division to match the standard mathematical definition of modulo.