Remainder

Remainder

The part left over after long division.

Long division example: 47 ÷ 11 = 4 remainder 3, shown with 44 subtracted from 47, leaving remainder 3.

Key Formula

a = bq + r, \quad 0 \le r < b

Where:

$a$ = The dividend — the number being divided
$b$ = The divisor — the number you divide by
$q$ = The quotient — how many whole times the divisor fits into the dividend
$r$ = The remainder — the part left over, always less than the divisor

Worked Example

Problem: Find the quotient and remainder when 157 is divided by 12.

Step 1: Set up the division equation. You need to find q and r such that:

157 = 12q + r, \quad 0 \le r < 12

Step 2: Determine how many whole times 12 fits into 157. Since 12 × 13 = 156 and 12 × 14 = 168, the quotient is 13.

q = 13

Step 3: Multiply the divisor by the quotient.

12 \times 13 = 156

Step 4: Subtract that product from the dividend to find the remainder.

r = 157 - 156 = 1

Step 5: Verify: check that the equation holds and that the remainder is less than the divisor.

12 \times 13 + 1 = 157 \quad \checkmark, \quad 1 < 12 \quad \checkmark

Answer: The quotient is 13 and the remainder is 1, so 157 ÷ 12 = 13 R 1.

Another Example

This example applies the concept of remainder to polynomial division rather than integer division, using the Remainder Theorem as a shortcut.

Problem: Find the remainder when the polynomial x³ + 2x² − 5x + 3 is divided by (x − 2).

Step 1: Use the Remainder Theorem: when a polynomial f(x) is divided by (x − c), the remainder equals f(c). Here c = 2.

r = f(2)

Step 2: Substitute x = 2 into the polynomial.

f(2) = (2)^3 + 2(2)^2 - 5(2) + 3

Step 3: Evaluate each term: 8 + 8 − 10 + 3.

f(2) = 8 + 8 - 10 + 3 = 9

Step 4: Verify by writing the division relationship. There exists some quotient polynomial q(x) such that:

x^3 + 2x^2 - 5x + 3 = (x - 2)\,q(x) + 9

Answer: The remainder is 9.

Frequently Asked Questions

What is the difference between a remainder and a decimal?

A remainder is the whole-number amount left over after integer division, while a decimal represents the same leftover as a fractional part of the divisor. For example, 17 ÷ 5 gives a remainder of 2, which corresponds to the decimal result 3.4 because 2/5 = 0.4. Remainders keep everything in whole numbers; decimals convert the leftover into a fraction of the divisor.

Can the remainder be zero?

Yes. A remainder of zero means the divisor divides the dividend exactly, with nothing left over. For instance, 20 ÷ 5 = 4 with a remainder of 0. When the remainder is zero, we say the dividend is divisible by the divisor.

Can the remainder be negative?

In standard division taught in school, the remainder is always a non-negative integer that is less than the divisor (0 ≤ r < b). Some advanced contexts allow negative remainders, but the most common convention requires the remainder to satisfy 0 ≤ r < |b|.

Remainder vs. Quotient

	Remainder	Quotient
Definition	The amount left over after division	The number of whole times the divisor fits into the dividend
Formula position	r in a = bq + r	q in a = bq + r
Constraint	Must satisfy 0 ≤ r < b	No upper bound; determined by a and b
Example (17 ÷ 5)	r = 2	q = 3

Why It Matters

Remainders appear throughout mathematics, from basic arithmetic and telling time (clocks use mod 12) to algebra and number theory. In algebra, the Remainder Theorem connects polynomial division to simple evaluation, which is a key tool for finding roots of polynomials. Understanding remainders also underpins modular arithmetic, which is essential in computer science, cryptography, and checking divisibility rules.

Common Mistakes

Mistake: Writing a remainder that is larger than or equal to the divisor.

Correction: The remainder must always be less than the divisor. If you get r ≥ b, the quotient is too small — increase q by 1 and recalculate r.

Mistake: Confusing the remainder with the decimal part of the answer.

Correction: The decimal 0.4 in 17 ÷ 5 = 3.4 is not the remainder. The remainder is 2, because 17 = 5 × 3 + 2. To convert, note that the decimal equals remainder ÷ divisor (2/5 = 0.4).

Related Terms

Quotient — The whole-number result paired with the remainder
Polynomial Long Division — Extends remainder concept to polynomial expressions
Synthetic Division — A shortcut method that produces a polynomial remainder
Remainder Theorem — States f(c) equals the remainder when dividing by (x − c)
Dividend — The number being divided in the division equation
Divisor — The number you divide by; remainder must be less than this
Long Division — The standard algorithm that produces a remainder
Factor — A divisor that gives a remainder of zero