Remainder Calculator
Calculate integer remainders with quotient-remainder identities, Euclidean remainders, truncating remainders, divisibility checks, and nearby multiples.
Last Updated: May 2026
Remainder
6
Quotient
7
Divisibility
Has remainder
Convention note
Same remainder
Remainder Inputs
Enter a whole-number dividend and divisor. The main remainder uses the Euclidean convention, while comparison rows show truncating and floor-division conventions.
The number being divided.
Any nonzero whole number.
Division Identities
| Identity | Calculation | Meaning |
|---|---|---|
| Euclidean identity | 55 = 7 x 7 + 6 | Uses a nonnegative remainder. |
| Remainder bounds | 0 <= 6 < 7 | Valid Euclidean remainder. |
| Truncating identity | 55 = 7 x 7 + 6 | Matches many programming remainder operators. |
| Floor identity | 55 = 7 x 7 + 6 | Uses floor division with the original divisor sign. |
Nearby Multiples
| Measure | Calculation | Value |
|---|---|---|
| Lower multiple | 7 x 7 | 49 |
| Next multiple | 49 + 7 | 56 |
| Distance from lower multiple | 55 - 49 | 6 |
| Distance to next multiple | 7 - 6 | 1 |
The remainder measures what is left after taking away whole divisor groups. In Euclidean division, that leftover is always nonnegative and smaller than the positive divisor.
Remainder Convention Notice
This calculator highlights the Euclidean remainder. Programming languages may use a truncating remainder operator with different negative-number behavior.
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How to Use the Remainder Calculator
Enter a whole-number dividend and a nonzero whole-number divisor. The calculator returns the remainder, quotient, divisibility status, and the division identity.
Use the comparison rows when negative numbers are involved. They show Euclidean, truncating, and floor-division conventions side by side.
Step 1: Enter the dividend
This is the number being divided, such as 55 or -55.
Step 2: Enter the divisor
Use any nonzero whole number.
Step 3: Read the remainder
The highlighted result uses the Euclidean convention with a nonnegative remainder.
Step 4: Check the identity
Confirm that dividend = divisor x quotient + remainder.
How This Remainder Calculator Works
The calculator parses the dividend and divisor as exact integers. It uses the absolute value of the divisor to compute the Euclidean remainder, so the result satisfies 0 <= r < |divisor|.
It also calculates truncating and floor-division identities. These are useful when comparing classroom arithmetic with programming-language remainder operators.
Nearby multiples show where the dividend sits between whole divisor groups. The distance from the lower multiple is the remainder.
Remainder Guide
Remainder Rules
| Rule | Formula | Meaning |
|---|---|---|
| Division identity | a = bq + r | Dividend equals divisor times quotient plus remainder. |
| Euclidean bounds | 0 <= r < |b| | The standard nonnegative remainder rule. |
| Divisibility | r = 0 | The dividend is evenly divisible by the divisor. |
| Truncating remainder | a % b in many languages | The sign can follow the dividend. |
| Floor remainder | a - b floor(a/b) | Uses floor division and can differ with negative divisors. |
Worked Examples
| Division | Identity | Remainder |
|---|---|---|
| 55 divided by 7 | 55 = 7 x 7 + 6 | 6 |
| 144 divided by 12 | 144 = 12 x 12 + 0 | 0 |
| -55 divided by 7 | -55 = 7 x -8 + 1 | 1 |
| 55 divided by -7 | 55 = 7 x 7 + 6 | 6 |
| 123456789 divided by 97 | 123456789 = 97 x 1272750 + 39 | 39 |
Remainders and Divisibility
Remainders are the foundation of divisibility tests. If the remainder is 0, the divisor fits exactly into the dividend. If the remainder is not 0, the remainder tells you how far the dividend is from the nearest lower multiple.
Keep the research moving with Modulo Calculator, Quotient Calculator, Division Calculator, and Floor Division Calculator.
Frequently Asked Questions
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Use Divisibility Test CalculatorSources & References
- 1.Khan Academy - Quotient Remainder Theorem(Accessed May 2026)
- 2.Wolfram MathWorld - Remainder(Accessed May 2026)
- 3.Wolfram MathWorld - Modulus(Accessed May 2026)