Modulo Calculator
Find a mod n, the remainder after division, with the quotient and steps, and see how negative numbers work in math and in programming languages.
Results
| Item | Value |
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Steps
| Step | Working |
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Quick Answer
a mod n is the remainder when a is divided by n. 47 mod 5 = 2, because 47 = 5 × 9 + 2. For negative numbers, math and Python use the floored rule (−7 mod 3 = 2), while JavaScript, C and Java use the truncated rule (−7 % 3 = −1).
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Table of Contents
How This Modulo Calculator Works
Enter the dividend a and the modulus n. The calculator divides a by n, takes the whole-number part of the quotient, and finds what is left over. It shows the floored remainder used in mathematics and Python, and the truncated remainder used by the % operator in JavaScript, C and Java, plus each quotient and the steps. Whole numbers of any length are handled exactly.
Quick explanation: mod is about what is left over.
- 17 mod 5 = 2: 5 goes into 17 three times with 2 left.
- Result range: for positive n, the floored result is from 0 to n − 1.
- Clock math: 15:00 is 3 o'clock because 15 mod 12 = 3.
How to Use the Modulo Calculator
- Enter the dividend a.
- Enter the modulus n.
- Press Calculate.
- Read a mod n and both remainder rules.
- Check the steps to see the division.
Core Formulas
- Definition: a = n × q + r
- Floored: q = ⌊a ÷ n⌋, r = a − n × q (r has the sign of n)
- Truncated: q = trunc(a ÷ n), r = a − n × q (r has the sign of a)
- Same answer: both rules agree when a and n are positive
Key Terms
- Modulo: the operation that finds the remainder after division.
- Modulus: the number you divide by, n.
- Dividend: the number being divided, a.
- Floor: rounding down to the next whole number.
- Congruent: two numbers are congruent mod n if they have the same remainder.
Worked Example
Find −7 mod 3.
- −7 ÷ 3 = −2.333...
- Floored: q = −3, r = −7 − 3 × (−3) = 2
- Truncated: q = −2, r = −7 − 3 × (−2) = −1, which is what −7 % 3 gives in JavaScript
Real-Life Examples
- Time: 100 hours from now is 100 mod 24 = 4 hours past this time, plus 4 days.
- Day of the week: 30 days from Monday is 30 mod 7 = 2 days later, Wednesday.
- Programming: cycle through a list with index mod length.
- Check digits: ISBN and other codes use mod 10 or mod 11 checks.
Modulo vs Remainder
- Modulo (floored): the result takes the sign of the divisor.
- Remainder (truncated): the result takes the sign of the dividend.
- For positive numbers: they are the same.
Modulo Examples
| a | n | Floored (math) | Truncated (JS %) |
|---|---|---|---|
| 17 | 5 | 2 | 2 |
| 47 | 5 | 2 | 2 |
| 100 | 7 | 2 | 2 |
| -7 | 3 | 2 | -1 |
| 7 | -3 | -2 | 1 |
| -7 | -3 | -1 | -1 |
| 24 | 6 | 0 | 0 |
| 365 | 7 | 1 | 1 |
Tips
- Even or odd: n mod 2 is 0 for even numbers and 1 for odd ones.
- Divisibility: a mod n = 0 means n divides a evenly.
- Positive results in code: use ((a % n) + n) % n to get the floored result.
- Large numbers: this calculator handles long whole numbers exactly.
Modulo is also the heart of modular arithmetic, where numbers wrap around like the hours on a clock. Two numbers are congruent mod n when they leave the same remainder, written 17 ≡ 2 (mod 5). This idea is used in calendars, hashing, cryptography and check digits on barcodes and bank numbers.
Common Mistakes to Avoid
- Assuming every language gives the same answer. Negative numbers differ.
- Using the quotient instead of the remainder. Mod gives what is left over.
- Dividing by zero. a mod 0 is undefined.
- Forgetting the range. With positive n, a floored result is always less than n.
- Rounding instead of flooring. The quotient must be floored or truncated, not rounded.
Frequently Asked Questions
It gives the remainder after dividing one number by another. 10 mod 3 = 1.
Divide a by n, take the whole number part, multiply it by n, and subtract from a.
2, because 47 = 5 × 9 + 2.
With the math rule, −7 mod 3 = 2. With JavaScript's %, −7 % 3 = −1.
Python floors the quotient, while JavaScript truncates it toward zero.
Yes. 7.5 mod 2 = 1.5.
To tell even (0) from odd (1) numbers.
For positive numbers, a mod n = a. For example, 3 mod 10 = 3.
No. Division gives the quotient, and modulo gives the remainder.
Math where numbers wrap around after reaching n, like hours on a clock.
Yes. With the floored rule, the result then has the same sign as n.
The fractional part of a, for positive a. 5.25 mod 1 = 0.25.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Modulo | Definitions, the floored and truncated rules, and how languages differ. | https://en.wikipedia.org/wiki/Modulo |
| Wikipedia, Modular arithmetic | Clock arithmetic and congruence. | https://en.wikipedia.org/wiki/Modular_arithmetic |