Return the standard nonnegative remainder after integer division. Input changes update both remainder and the supporting steps.
Use Dividend as the first Modulo checkpoint. Confirm Positive modulus, then anticipate Remainder. Repeat Modulo without reading the prior answer. If Positive modulus differs, preserve both Modulo versions and both values of Remainder.
−17 mod 5 is 3 under the nonnegative convention because −17 = (−4) × 5 + 3. The remainder lies in the required range 0 through 4.
Modulo arithmetic groups integers by their remainders after division by a positive modulus. Two integers are congruent modulo n when their difference is divisible by n. For a connected concept in Modulo, see divisibility structure.
Find integers q and r satisfying a = qn + r with 0 ≤ r < n. For a negative dividend, choose the floor-style quotient that leaves a nonnegative remainder.
Programming languages differ in how the % operator handles negative dividends. Some return a signed remainder such as −2; this page normalizes the result to the usual mathematical range 0 ≤ r < n.
Start the Modulo review with Dividend. Compare Dividend with its source, then test Positive modulus in a second Modulo run without changing the first Modulo case.
The Modulo meaning depends on Dividend. The Modulo meaning also depends on Positive modulus. Carry those Modulo roles into any later Modulo work.
Clock arithmetic, repeating schedules, array indexing, checksums, and divisibility patterns all use remainders. Modulo turns an unlimited integer line into a repeating cycle of residue classes. A related application of Modulo is integer factors.
The quotient is not displayed because many different quotients can accompany congruent integers while the residue remains the same. For modulus 5, values such as 3, 8, −2, and 103 all occupy the residue class represented by 3.
Modular addition and multiplication can reduce operands before the operation. Replacing large integers by their residues keeps the arithmetic small and does not change the final residue, a property frequently used in divisibility checks and repeating-cycle problems.
For positive inputs they usually coincide; negative-input conventions can differ.
It gives a clear standard residue range from 0 through n − 1.
Always 0, because every integer is divisible by 1.
Yes. An integer is even exactly when n mod 2 equals 0.