Modulo Calculator

Modulo arithmetic describes what remains after one number is divided by another. In the expression a mod b, a is the dividend and b is the modulus or divisor. The result is the remainder associated with division by b. For example, 17 divided by 5 has an integer quotient of 3 and leaves 2, so 17 mod 5 equals 2. This simple operation appears throughout elementary arithmetic, number theory, computer science, cryptography, scheduling, and data structures. This modulo calculator reports both the Euclidean remainder and the corresponding quotient. It chooses a remainder from zero up to, but not including, the absolute value of the divisor. That convention keeps the answer predictable for negative dividends. For example, negative 13 mod 5 is 2 because negative 13 equals 5 times negative 3 plus 2. Some programming languages use a remainder operator that preserves the sign of the dividend instead, so their result for the same inputs may be negative 3. Knowing which convention is in use matters when translating a mathematical formula into code. Modulo is especially useful for repeating cycles. Clock arithmetic is a familiar example: adding five hours to 10 o'clock gives 15, and 15 mod 12 gives 3 o'clock. Weekday calculations use modulus 7, circular array indexes use the array length, and parity checks use modulus 2. In programming, a zero remainder can test divisibility, distribute records among buckets, wrap positions around a collection, or select every nth item. In number theory, congruence says two integers are equivalent modulo b when they have the same remainder after division by b. The divisor cannot be zero because division by zero has no defined quotient or remainder. Decimal inputs are accepted for convenient remainder calculations, although modular arithmetic in proofs and cryptographic algorithms normally uses integers. When a problem asks for an integer congruence, enter whole numbers and interpret the displayed nonnegative remainder as the canonical representative of its congruence class. The calculator performs the operation immediately and shows the division identity, making it easy to verify the result without doing repeated subtraction by hand.