Modulo Calculator – Guide & Formulas
Calculate the modulo (remainder) of any division with our free online calculator. Enter two numbers to find a mod b instantly with step-by-step work.
Calculate the modulo (remainder) of any division with our free online Modulo Calculator. Enter two numbers to find a mod b instantly with step-by-step division work.
Key Takeaway
Use the free Modulo Calculator to calculate the modulo (remainder) of any division with our free online calculator. enter two numbers to find a mod b instantly with step-by-step work. Get instant results with step-by-step explanations.
How to Use the Modulo Calculator
- Step 1: Enter the dividend a (the number being divided)
- Step 2: Enter the divisor b (the number to divide by, must be greater than 0)
- Step 3: Click Calculate to find a mod b (the remainder when a is divided by b)
- Step 4: Review the step-by-step division showing quotient and remainder
The Formula
Variable Definitions
- a: The dividend (the number being divided)
- b: The divisor (the number to divide by; must be > 0)
- mod: The modulo operator — returns the remainder of division
- ⌊x⌋: The floor function — rounds x down to the nearest integer
- Quotient: The integer result of dividing a by b (how many times b fits into a)
Calculating 17 mod 5
Find the remainder when 17 is divided by 5
- Divide 17 by 5: 17 ÷ 5 = 3.4, so the quotient (floor) is 3.
- Multiply quotient by divisor: 3 × 5 = 15.
- Subtract from dividend: 17 − 15 = 2.
- Result: 17 mod 5 = 2. Verify: 17 = 5 × 3 + 2 ✓
Frequently Asked Questions
What is the modulo operation?
The modulo operation finds the remainder after dividing one number by another. For example, 17 mod 5 = 2 because 17 divided by 5 equals 3 with a remainder of 2.
How do I calculate a mod b?
Divide a by b, take the integer part of the quotient, multiply by b, and subtract from a. The result is a mod b. Alternatively, repeatedly subtract b from a until the result is less than b.
What is the difference between modulo and remainder?
In most cases they are the same. However, some programming languages define remainder differently for negative numbers. This calculator uses the mathematical definition where a mod b is always non-negative when b > 0.
Can the modulo result be zero?
Yes, a mod b = 0 when a is exactly divisible by b (b divides a evenly). For example, 12 mod 4 = 0 because 12 ÷ 4 = 3 with no remainder.
What happens if I enter 0 as the divisor?
Division by zero is undefined in mathematics. The modulo operation requires the divisor to be greater than 0. This calculator will show an error if you enter 0 as the divisor.
What are practical uses of modulo?
Modulo is used in cryptography, hash functions, clock arithmetic, checking divisibility, generating random numbers, programming (array indexing), and many areas of computer science and mathematics.
How does modulo work with negative numbers?
When the dividend is negative, the modulo result depends on the convention used. This calculator returns a non-negative result: (−7) mod 3 = 2, because −7 = 3 × (−3) + 2.
What is modular arithmetic?
Modular arithmetic is a system of arithmetic where numbers wrap around after reaching a certain value (the modulus). For example, in mod 12 arithmetic (like a clock), 10 + 5 = 3, not 15.
Is modulo the same as percent?
No, modulo finds the remainder of division, while percent means "per hundred." However, the word "modulo" is sometimes abbreviated as "mod" which can be confused with the percent sign in some contexts.
What is the largest modulo result possible?
The result of a mod b is always in the range [0, b−1]. So the largest possible result for a given divisor b is b−1. For example, a mod 10 can be at most 9.