What is the remainder in integer division?

Given integers a (dividend) and b (divisor, non-zero), the remainder is the integer r that appears in the identity a = b · q + r, where q is the quotient and, in Euclidean division, the remainder r satisfies 0 ≤ r < |b|. This r measures what is left over after dividing a by b as evenly as possible.

What is the difference between remainder and modulo?

In pure mathematics, the modulo operation and the Euclidean remainder coincide: both give the unique r with 0 ≤ r < |b| such that a = b · q + r. In programming languages, however, operators often implement a truncated-division remainder where the sign of r can follow the dividend. This calculator shows both the Euclidean remainder and a typical programming-style remainder so you can compare them, especially for negative inputs.

How does this calculator handle negative numbers?

For the Euclidean remainder, the calculator always returns a remainder r satisfying 0 ≤ r < |b|, even when a is negative. It also shows a truncated-division remainder based on rounding the quotient toward zero, which is similar to the behavior of the % operator in many languages. The two values coincide for non-negative dividends but differ for negative dividends.

What input range is supported?

The calculator expects integer inputs in a range where standard double-precision arithmetic is exact for integer values, typically around |a|, |b| ≤ 10^15. For extremely large integers, consider using an arbitrary-precision environment or a computer algebra system.

Where are remainder and modulo used in practice?

Remainders appear throughout mathematics and computing: in clock arithmetic, cyclic patterns, hashing functions, pseudo-random number generators, cryptography, checksums, and algorithms such as the Euclidean algorithm for computing greatest common divisors.

Remainder Calculator – Integer Division, Quotient and Remainder

Q: What is the difference between remainder and modulo?

In pure mathematics, the modulo operation and the Euclidean remainder coincide: both give the unique r with 0 ≤ r < |b| such that a = b · q + r. In programming languages, however, operators often implement a truncated-division remainder where the sign of r can follow the dividend. This calculator shows both the Euclidean remainder and a typical programming-style remainder so you can compare them, especially for negative inputs.

Q: How does this calculator handle negative numbers?

For the Euclidean remainder, the calculator always returns a remainder r satisfying 0 ≤ r < |b|, even when a is negative. It also shows a truncated-division remainder based on rounding the quotient toward zero, which is similar to the behavior of the % operator in many languages. The two values coincide for non-negative dividends but differ for negative dividends.

Q: What input range is supported?

The calculator expects integer inputs in a range where standard double-precision arithmetic is exact for integer values, typically around |a|, |b| ≤ 10^15. For extremely large integers, consider using an arbitrary-precision environment or a computer algebra system.

Q: Where are remainder and modulo used in practice?

Remainders appear throughout mathematics and computing: in clock arithmetic, cyclic patterns, hashing functions, pseudo-random number generators, cryptography, checksums, and algorithms such as the Euclidean algorithm for computing greatest common divisors.

Enter an integer dividend and divisor to compute the quotient and remainder of the division, verify the identity a = b·q + r, and compare the Euclidean remainder (always non-negative) with a typical programming-style remainder for negative numbers.

Mathematical definition of remainder

Let \(a\) and \(b\) be integers with \(b \ne 0\). A quotient–remainder pair \((q, r)\) satisfies

\[ a = b \cdot q + r. \]

In Euclidean division, which is the standard in number theory, we additionally require

\[ 0 \le r < |b|. \]

Under this condition, the integers \(q\) and \(r\) are unique for any given pair \(a, b\). This is the convention used to define the modulo operation and arithmetic in \(\mathbb{Z}/n\mathbb{Z}\).

Euclidean remainder formula (conceptual)

For a fixed positive divisor \(b > 0\), the Euclidean quotient is \[ q = \left\lfloor \frac{a}{b} \right\rfloor, \] and the remainder is \[ r = a - b \left\lfloor \frac{a}{b} \right\rfloor, \] so that \(0 \le r < b\).

For negative divisors, we can always work with \(|b|\) and then adjust the quotient accordingly.

Remainder vs modulo in programming languages

Many programming languages (C, C++, Java, JavaScript, etc.) define a remainder operator like a % b using a quotient rounded toward zero. That is, they take

\[ q_{\text{trunc}} = \operatorname{trunc}\!\left(\frac{a}{b}\right), \quad r_{\text{trunc}} = a - b \, q_{\text{trunc}}, \]

where trunc drops the fractional part. This makes \(r_{\text{trunc}}\) have the same sign as the dividend \(a\), and it can be negative.

In contrast, the Euclidean remainder \(r\) is defined so that \(0 \le r < |b|\), which is more convenient for mathematical proofs and modular arithmetic.

Worked examples

a	b	Euclidean q, r	Programming q_trunc, r_trunc	Notes
17	5	q = 3, r = 2	q = 3, r = 2	Positive case: both conventions coincide.
-17	5	q = -4, r = 3	q = -3, r = -2	Euclidean r is non-negative; truncated r follows the sign of a.
17	-5	q = -3, r = 2 (since \|b\| = 5)	q = -3, r = 2	For this pair, both conventions happen to give the same remainder.
-17	-5	q = 4, r = 3 (with \|b\| = 5)	q = 3, r = -2	Again, Euclidean r is in [0, \|b\|), truncated r is negative.

How to interpret the result in practice

Use the Euclidean remainder for mathematical work, modular arithmetic, congruences, and anything involving \(\bmod\) in textbooks.
Use the programming-style remainder when you specifically need to mirror the behavior of a concrete language operator (for example, debugging code or porting algorithms).
When mixing math and code, always state which convention you are using to avoid off-by-one or sign errors.

Typical applications of remainders

Clock arithmetic (e.g., converting 27 hours into 1:00 using modulo 24).
Detecting even/odd numbers (checking if a number is congruent to 0 or 1 modulo 2).
Distributing items into cycles or repeating patterns.
Hash functions and hashing to array indices.
Cryptographic algorithms and checksums.
The Euclidean algorithm for greatest common divisors (GCD).

Remainder Calculator

Compute quotient and remainder of a ÷ b

Results

Euclidean division (number theory standard)

Truncated division (programming-style remainder)

Comparison