A modulo calculator finds the remainder class associated with dividing one number by another. For positive integers, the result is the ordinary remainder. With negative values, however, mathematical modulo and programming-language remainder rules can produce different answers.
What Is Modulo?
Modulo, written as a mod b, describes the remainder associated with dividing a by b. For example, 17 mod 5 = 2 because:
Here, 17 is the dividend, 5 is the divisor or modulus, 3 is the quotient, and 2 is the modulo result.
Modulo Formula
For a positive modulus b, mathematical modulo can be written as:
This produces the least non-negative result for a positive modulus:
How to Calculate Modulo Step by Step
- Divide the dividend by the divisor.
- Find the appropriate integer quotient.
- Multiply the divisor by that quotient.
- Subtract the product from the dividend.
- The value left is the mathematical modulo result.
- Verify with a = bq + r.
Example: 47 mod 8
47 ÷ 8 = 5.875
Integer quotient = 5
8 × 5 = 40
47 − 40 = 7
47 mod 8 = 7
Verification: 47 = 8 × 5 + 7
Modulo vs Remainder
For positive values, mathematical modulo and programming remainder usually match. The difference becomes important with negative values.
| Expression | Mathematical Modulo | JavaScript Remainder |
|---|---|---|
| 17 and 5 | 17 mod 5 = 2 | 17 % 5 = 2 |
| -7 and 5 | -7 mod 5 = 3 | -7 % 5 = -2 |
| 7 and 5 | 7 mod 5 = 2 | 7 % 5 = 2 |
This calculator shows both results so you can clearly distinguish mathematical modulo from JavaScript-style remainder when negative values are involved.
Negative Modulo Example
-7 mod 5
floor(-7 / 5) = -2
-7 − 5(-2) = 3
Mathematical modulo: -7 mod 5 = 3
JavaScript remainder: -7 % 5 = -2
Modulo in Programming
The percent sign % is commonly used as a remainder operator in programming languages. It is useful for repeating cycles, index wrapping, parity checks, and scheduling patterns.
Even and Odd Numbers
If n % 2 = 0, n is even. If n % 2 is not 0, n is odd.
Circular Indexing
For an array of length 7, an index can be wrapped with:
wrappedIndex = index % 7
Clock Arithmetic
Modulo is often compared with a clock because values wrap around after a fixed cycle.
23 mod 12 = 11
17 mod 12 = 5
36 mod 12 = 0
Common Modulo Examples
| Calculation | Result | Use |
|---|---|---|
| 42 mod 2 | 0 | Even/odd check |
| 365 mod 7 | 1 | Weekly cycles |
| 28 mod 12 | 4 | Clock arithmetic |
| 5 mod 10 | 5 | Divisor larger than dividend |
Common Modulo Mistakes
1. Treating % and mathematical modulo as identical
They often match for positive values but may differ for negative values.
2. Dividing by zero
Modulo by zero is undefined because division by zero is undefined.
3. Confusing quotient with remainder
Division gives the quotient. Modulo focuses on the remainder class.
4. Assuming decimal arithmetic is always exact
Browser calculations use floating-point numbers, so decimal inputs can produce tiny precision differences.
5. Using very large integers with normal JavaScript numbers
Extremely large integers may lose exact precision when represented as ordinary JavaScript Number values.
Quick Modulo Reference
- mod 2: Even/odd pattern
- mod 7: Weekly cycles
- mod 10: Last decimal digit for non-negative integers
- mod 12: Clock cycles
- mod 24: Daily hour cycles
- mod 60: Minutes and seconds cycles
Related Calculators
How This Modulo Calculator Works
The calculator computes two related results when needed. The programming remainder follows JavaScript's % operator. The mathematical modulo uses the least non-negative residue for a positive modulus.
For ordinary positive inputs, both values are the same. For negative values, they may differ. The calculator also shows the quotient, step-by-step calculation, and the verification identity a = bq + r.
Prepared by the Utilixea Editorial Team.
Last updated: July 21, 2026.
Modulo Calculator FAQs
A modulo calculator finds a mod b, the remainder class associated with dividing a by b.
For positive values they usually match. With negative numbers, mathematical modulo and programming remainder conventions can produce different results.
Using the usual least non-negative mathematical convention, -7 mod 5 = 3.
JavaScript's % operator returns the remainder after truncated division, so the sign follows the dividend in this case.
Modulo by zero is undefined because division by zero is undefined.
Programming languages often allow decimal remainder calculations, but floating-point precision can affect the result. Integer inputs are recommended for standard mathematical modulo.
For positive integers, the modulo result is usually the dividend itself. For example, 5 mod 10 = 5.
Use the identity a = bq + r. For example, 47 mod 8 = 7 because 47 = 8 × 5 + 7.
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