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Euclidean Division

Tool to make an euclidean division automatically. Euclidean division is an arithmetical operation which associates to two numbers: the dividend and the divisor, two others numbers resulting from the division operation : the quotient and the remainder.

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Euclidean Division -

Tag(s) : Arithmetics

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# Euclidean Division

## Long Division A/B

Tool to make an euclidean division automatically. Euclidean division is an arithmetical operation which associates to two numbers: the dividend and the divisor, two others numbers resulting from the division operation : the quotient and the remainder.

### How to calculate the quotient of the euclidean division?

Quotient $q$ is the integer part of the division a/b where a is the dividend and b the divisor.

Example: In the division (or fraction) $43/21 = 2.047619\dots$, the quotient $q = 2$ (the integer part)

### How to calculate the remainder of the euclidean division?

The remainder $r$ is the result of $$r = a - q \times b$$

Example: In the division $43/21$, the quotient equals $2$ and the remainder equals $43 - 21 \times 2 = 1$, indeed $43 = 2 \times 21 + 1$.

### How to make the remainder a positive value?

To get a positive remainder, take for quotient $q-1$, the remainder is then augmented by $b$.

Example: If $a = 15, b = 4$, then is is possible to use $q = 4, r = -1$ (negative) because $b \times q + r = 4 \times 4 - 1 = 15 = a$ but to get a positive remainder, take $q = ( 4 - 1 ) = 3$ and then have $r = 3$ (positive) and $b \times q + r = 4 \times 3 + 3 = 15 = a$

### How to calculate a division with a decimal number?

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.

Example: 1.2/3 is equivalent to having 12/30

### What is the difference between decimal division and Euclidean division?

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

### Why the name Euclidean Division?

The name comes from Euclid, a mathematician.

### What are the limits of this software?

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

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