Modulo Calculator
Calculate the remainder of any division operation instantly.
The number being divided.
The divisor cannot be zero.
Modulo result
17 mod 5 = 2
Remainder after division.
Quotient
3
How many full times the divisor fits.
Dividend
17
Divisor
5
Mode
Euclidean modulo
For a positive modulus, the remainder is kept non-negative.
Equation form
17 = 5 × 3 + 2
Congruence form
17 ≡ 2 (mod 5)
Modulo Formulas and Rules
Modulo / remainder
a mod b = r
Division relationship
a = bq + r
Euclidean quotient
q = floor(a ÷ b), when b is positive
Remainder range
0 ≤ r < b, when b is positive in Euclidean modulo
Congruence
a ≡ r (mod b)
Example
17 = 5 × 3 + 2, so 17 mod 5 = 2
Variable Explanations
a
Dividend, or number being divided.
b
Divisor, or modulus.
q
Quotient, or full division count.
r
Remainder, or modulo result.
mod
Modulo operation.
Congruence
Two numbers have the same remainder after division by the modulus.
Rule
The divisor cannot be zero.
What Modulo Means
Remainder after division
Modulo finds what is left after dividing.
Closely related to division
It uses quotient and remainder.
Repeating cycles
Modulo is useful when values wrap around.
Even or odd
10 mod 2 = 0, so 10 is even.
Divisibility
14 mod 7 = 0, so 14 is divisible by 7.
Programming use
Modulo appears in clocks, calendars, hashing, and array indexing.
Worked Examples
17 mod 5
Division statement: 17 = 5 × 3 + 2
Quotient: 3
Remainder: 2
5 fits into 17 three times with 2 left over.
20 mod 4
Division statement: 20 = 4 × 5 + 0
Quotient: 5
Remainder: 0
A zero remainder means 20 is divisible by 4.
23 mod 10
Division statement: 23 = 10 × 2 + 3
Quotient: 2
Remainder: 3
This is useful for extracting the last digit of a positive integer.
7 mod 3
Division statement: 7 = 3 × 2 + 1
Quotient: 2
Remainder: 1
The remainder after dividing by 3 is 1.
Even/odd with mod 2
Division statement: 11 = 2 × 5 + 1
Quotient: 5
Remainder: 1
A remainder of 1 means 11 is odd.
Clock example
Division statement: 14 mod 12
Quotient: 1
Remainder: 2
On a 12-hour cycle, 14 wraps to 2.
Array index example
Division statement: index mod arrayLength
Quotient: varies
Remainder: wrapped index
Modulo keeps indexes inside a repeating range.
Negative example
Division statement: -7 mod 3
Quotient: -3 or -2 depending on convention
Remainder: 2 in Euclidean mode
Negative modulo behavior depends on the chosen convention.
Modulo in Programming and Number Theory
Even/odd checks
n % 2 checks whether a number is even or odd.
Divisibility
n % divisor = 0 means exact divisibility.
Wrapping indexes
index % arrayLength keeps indexes inside an array.
Clock arithmetic
hour % 12 wraps time around a 12-hour clock.
Cyclic schedules
Modulo helps repeat schedules or rotations.
Number theory
Congruences compare remainders under the same modulus.
Negative Numbers and Modulo Behavior
Euclidean modulo
Euclidean modulo keeps the remainder non-negative when the divisor is positive. This is often preferred in math explanations.
Programming remainders
Some programming languages return a remainder with the same sign as the dividend. This calculator lets you compare Euclidean and JavaScript-style behavior.
Common Modulo Mistakes
Understanding Your Results
Modulo result
The remainder left after division.
Quotient
How many full times the divisor fits into the dividend.
Equation form
Shows dividend as divisor times quotient plus remainder.
Congruence form
Shows the equivalent remainder under the modulus.
Zero remainder
Means the dividend is divisible by the divisor.
Frequently Asked Questions
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