About

Integer division constitutes the foundation of modular arithmetic, computer science algorithms, and cryptographic hashing functions. Unlike standard floating-point division, Euclidean division segregates the result into an integer quotient and a strictly positive remainder. Precision in this operation is critical for developers handling array indexing, clock arithmetic, or distributing items into fixed containers. This tool visualizes the grouping process, providing immediate clarity on how the remainder is derived. It handles negative integers according to Euclidean definitions where the remainder is always non-negative, while also providing standard programming language syntax outputs.

Formulas

The core mechanism of Euclidean division ensures that for any integer dividend a and non-zero integer divisor b, there exist unique integers q (quotient) and r (remainder) satisfying:

Euclideana = bq + r0 ≤ r < |b|

In programming, the modulo operator behavior varies with negative numbers. This tool calculates the mathematical definition.

Reference Data

Dividend (a)	Divisor (n)	Quotient (q)	Remainder (r)	Equation	Mod Logic
10	3	3	1	10 = 3×3 + 1	10 mod 3 = 1
14	4	3	2	14 = 4×3 + 2	14 mod 4 = 2
25	5	5	0	25 = 5×5 + 0	25 mod 5 = 0
7	2	3	1	7 = 2×3 + 1	7 mod 2 = 1
-10	3	-4	2	-10 = 3×-4 + 2	Euclidean def.
100	7	14	2	100 = 7×14 + 2	Cycle logic
1	2	0	1	1 = 2×0 + 1	Base case
60	12	5	0	60 = 12×5 + 0	Time (min)

Frequently Asked Questions

Programming languages handle negative operands differently. Python's modulo operator yields a result with the same sign as the divisor, effectively implementing "floored division". C++ (prior to C++11) and many standard processors use "truncated division", where the remainder takes the sign of the dividend. This tool displays the strict Euclidean result where the remainder is always positive.

Division by zero is undefined in arithmetic. In computing, this throws a runtime exception or results in Infinity. This tool restricts the divisor input to non-zero integers to maintain mathematical validity.

The quotient is the integer part of the division result. It represents how many full times the divisor "fits" into the dividend. Any remaining value that is less than the absolute value of the divisor becomes the remainder.

Yes, the calculation logic supports integers up to the maximum safe integer limit of the browser environment (approximately 15 digits). However, the visual grouping feature (dots) is automatically disabled for dividends exceeding 500 to prevent rendering lag.