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Yes. Remainder and quotient can be the same depending on the dividend and divisor. For example, 12÷5 gives 2 as the quotient and the remainder.
Does an even divisor and odd dividend always produce a remainder?
An even divisor with an odd dividend will always generate a remainder.
Does 297 divided by 3 generate any remainder?
The sum of the digits 297 is 18 (2+9+7), divisible by 3. Therefore, 297 is divisible by 3 without leaving any remainder.
Can we find the dividend if the divisor, Quotient, and remainder are given?
Yes. You can find the dividend using the formula: Dividend = Remainder + (Divisor x Quotient).
Remainder: Formula, Properties, and Examples
Remainder is the leftover number after division when a number isn’t divided evenly. Use the formula: Remainder = Dividend − (Divisor × Quotient). It’s always less than the divisor. Remainders appear in real-life sharing problems and help convert improper fractions to mixed numbers. Mastering this improves accuracy in division and builds a strong math foundation.
Shivam Singh18 Aug, 2025
Remainder: When we divide one number by another, the division sometimes does not end perfectly, and we are left with something extra. This extra part is called the remainder. When distributing things equally among a group or calculating changes after a purchase, the concept of remainder comes into practice.
It's an important mathematical concept that helps solve division-related problems easily and accurately. We will discuss the remainder in the division here, as well as its properties and formulas to help young learners understand and apply this concept in different contexts.
As the name suggests, the remainder in division is what 'remains' after completing the division process. In mathematical terms, a reminder is the number that is left over in division when the dividend is not perfectly divisible by the divisor.
Let's simplify the remainder definition with an example.
If we divide 27 candies equally among 6 children, each child will get 4 candies, and 3 candies will be left after the distribution. Here, 27 is not perfectly divisible by 6, so a remainder is generated.
In this example,
27 is the dividend (the number being divided)
6 is the divisor (the number by which we divide the dividend)
4 is the quotient (the whole number result)
3 is the remainder (the remaining part after division)
Formula for Remainder
The formula for remainder can be derived from the remainder definition. We get the remainder if we multiply the quotient by the divisor and subtract the product from the dividend.
Therefore, we can write the formula for the remainder as follows:
Remainder = Dividend – (Divisor x Quotient)
We can also write the formula as below:
R = D – (d x q)
Where,
D = Dividend
d = Divisor
q = Quotient
R = Remainder
For example, if D = 52, d = 16, q = 3, then we can calculate the remainder as:
The properties of remainder are useful for solving division-related problems efficiently and accurately. Look at the remainder properties as follows:
The remainder is always less than the divisor because otherwise, you can continue the division, meaning the process is incomplete. For example, 32÷5 gives a remainder of 2, less than 5.
The remainder is zero when the dividend is completely divisible by the divisor. For example, 54 is divisible by 9, so 54÷9 gives a remainder of zero.
The remainder in division can be greater, lesser, or equal to the quotient.
The remainder is always taken as a non-negative number when dividing a negative number by a positive number or vice versa.
Remainder Theorem Formula
The remainder theorem formula is related to the division of polynomial functions. It states that if a polynomial f(x) is divided by (x-a), then the remainder will be f(a).
Here,
f(x) represents the polynomial you are dividing.
x – a is the divisor, where 'a' is a constant.
f(a) gives the value of the polynomial f(x) when x is replaced with 'a'.
In simpler terms, if we consider x = a, and substitute this value in the polynomial f(x), we will get a value which is the remainder when f(x) is divided by (x -a).
For example,
Let, f(x) = x2 - 5x + 12 which is divisible by (x -3)
Remainders are an important component in expressing mixed fractions. When we denote a fraction with a value greater than 1, we divide the numerator by the denominator to get the mixed fraction equivalent.
This mixed fraction has a whole number part and a fractional part. The whole number part indicates the quotient, the denominator of the fractional part denotes the divisor, and the numerator of the fractional part denotes the remainder.
For example,
The fraction 7/3 indicates division of 7 by 3 which gives 2 as the quotient and 1 as the remainder. So, we express 7/3 as 21/3.
The remainder is an important part of the division method and closely relates to the dividend, divisor, and quotient. Understanding the remainder formula and properties helps your child to make quick and accurate calculations and excel in school assignments.
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