What is Quotient and Remainder?

A quotient can be an integer in the case of Euclidean division, a fraction, or a ratio in the case of a proper division. The quotient is defined as the quantity produced after dividing two numbers, or it is also defined as the proper ratio of two numbers. The quotient has significant uses in all the branches of mathematics.

Integer Quotient

The integer quantity produced after dividing two numbers is referred to as the integer quotient. It is also defined as the greatest whole number of times a divisor can be subtracted from the dividend before the remainder becomes negative. For example, when we divide 10 by 5, we get 2 as an integer quotient. Here 10 is referred to as dividend or numerator, and 5 is referred to as divisor or denominator.

More Examples

Here we can subtract 4 from 20, 5 times before the remainder becomes negative. Hence the quotient is 5.

Rational Quotient

The rational quantity in the form of (P / Q) produced by dividing two numbers is referred to as the rational quotient. A rational quotient is produced when the numerator is not divisible by the denominator. For example, when we divide 10 by 6, we get 5 / 3 or 1.67 as quotient. More examples:

Remainder

The remainder is divided into the following sections, such as:

Least Positive Remainder

The remainder is defined as the least positive value that remains after subtracting the divisor from the maximum dividend number of times. The range of the remainder 0 <= r < divisor, here r is remainder. The range denotes that the reminder can’t be negative and equal to or greater than the divisor. We can write the dividend in the form of q and r. General equation to denote a dividend:

D = d * q + r, Here D is the dividend, d is the divisor, q is the quotient, and r is the remainder.

Least Absolute Remainder

An absolute remainder is the largest negative value obtained after subtracting the divisor from the maximum dividend number of times. It can also be obtained by subtracting the divisor from the least positive remainder.

Least positive remainder = divisor + Least Absolute remainder

For Example

When we divide 38 by 5, we get 7 as the quotient and 3 as the remainder,

38 = 5 * 7 + 3, where 3 is the least positive remainder

And, 38 = 5 * 8 + (-2), where -2 is the least absolute remainder.

The above definitions are valid for the negative values too. For example:

When we divide 38 by -5, we get -7 as the quotient and 3 as the least positive remainder,

38 = (-5) * (-7) + 3

And 38 = (-5) * (-8) + (-2), where -2 is the least absolute remainder.

In the case of floating-point numbers, it is unnecessary to have a remainder in the result. The dividend can be denoted as the product of the quotient and the divisor (D = q * d).

For example, when we divide 5 by 2, we get 2.5 (a floating-point) as the quotient, and here remainder is 0.

5 = 2.5 * 2 or 5 = 2 * 2.5 + 0

When we divide 34.5 by 4.9, we get 7 as the quotient. 34.3 = 7 * 4.9