Long Division with Remainder
For decimal numbers (e.g. 112.45 ÷ 56.7), use : Decimal division.
For polynomials, e.g. (x2 + 1) ÷ (x - 1), use : Polynomial euclidean division.
Remainder = 71
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Euclidean division of two positive integers
Doing the integer division of a positive integer (the dividend) by another non-zero positive integer (the divisor) consists on finding two integers called the quotient and the remainder that verifies the equality,
dividend = (quotient × divisor) + remainder
the remainder is an integer less than the divisor.
Example: 17 ÷ 5 = 3 remainder 2
Euclidean division of two relative integers
The above definition can be generalized to two integers that might be positive or negative (relative integers).
If a is the the dividend and b the divisor,
then there exist unique two integers q (quotient) and r (remainder) such that,
`a = b. q + r` and `0 <= r < |b|`
- Case of positive integers:
23 ÷ 4 = 5 remainder 3
56 ÷ 7 = 8 remainder 0
- Cases of relative integers
-23 ÷ 5 = -5 remainder 2
-65 ÷ 3 = -22 remainder 1
45 ÷ -4 = -11 remainder 1
-26 ÷ -7 = 4 remainder 2
- Special cases:
If the dividend is 0 then the quotient and the remainder are equal to 0.
0 ÷ 3 = 0 remainder 0
If the dividend is equal to the dividend then the quotient is equal to 1 and the remainder is 0.
24 ÷ 24 = 1 remainder 0
If the dividend is a multiple of the dividend then the remainder is equal to 0.
9 ÷ 3 = 3 remainder 0
Integer division and modulo
For two given integers a and b, the remainder of the Euclidean division of a by b is congruent to a modulo b,
`a\equiv r\mod b`
r being the remainder of the Euclidean division of a by b.
Here is a program that computes the quotient and the remainder of the Euclidean division of two integer numbers a (dividend) and b (divisor).
Note that in python, the remainder of the integer division may be negative.
In the above calculator, the remainder is always positive (or zero) which guarantees its uniqueness.
def euclidean_division(a, b): #quotient = a//b, remainder = a % b return (a//b, a%b)