Euclidean Division

Euclidean division is an arithmetical operation which associates to two (integer) numbers: the dividend and the divisor (not 0), two others numbers resulting from the division operation: the quotient and the remainder.

It can be calculated by hand with several steps (long division) or directly using a calculator.

Quotient $ q $ is the integer part of the division a/b (ignoring the resulting decimal part) where a is the dividend and b the divisor.

The remainder $ r $ is the result of $$ r = a - q \times b $$. It represents what is left after the largest possible divisor multiple has been subtracted from the dividend (the remainder after the sharing).

To find the dividend of Euclidean division, you can multiply the quotient by the divisor and add the remainder to it. The dividend $ a $ is then calculated from the other values $$ a = b \times q + r $$

The divisor $ b $ is calculated from the other values $$ b = \frac{a - r}{q} $$

The quotient $ q $ is calculated from the other values $$ q = \frac{a - r}{b} $$

In mathematics, the remainder of Euclidean division can be positive or negative, but it is commonly agreed to take the positive remainder. If the remainder obtained is negative, you can add the divisor to it to make it positive by subtracting 1 from the quotient. To obtain a positive remainder, take as quotient $ q-1 $, the remainder is then increased by $ b $.

An Euclidean division is an entire division, it is not supposed to have decimal places.

However, if the divisor or the dividend has decimal places, it is possible to multiply by 10 the dividend and the divisor as many times as necessary to make it disappear.

If the dividend is less than the divisor then the result of the division is 0 and the remainder is the dividend.

If the division stops at integer numbers (before the decimal point) then it is a Euclidean division, otherwise it is a decimal division.

A modulo (mod) calculation is a division that is only interested in the remainder.

The name euclidean division comes from Euclid, a mathematician.

This software is not limited, it can calculated with any number, including big numbers with arbitrary precision.