The remainder is the amount 'left over' after performing a division operation when one number cannot be perfectly divided by another. For example, 10 divided by 3 is 3 with a remainder of 1.

The remainder is calculated using the formula: Dividend = (Divisor × Quotient) + Remainder. You find the largest whole number quotient that, when multiplied by the divisor, is less than or equal to the dividend. The difference is the remainder.

In standard arithmetic, the remainder is typically non-negative and less than the absolute value of the divisor. However, some programming languages and mathematical contexts define the modulo operation (often related to remainder) differently, which can result in negative remainders when negative numbers are involved. Our calculator provides the standard arithmetic remainder.

While often used interchangeably, 'remainder' and 'modulo' can have subtle differences, especially when dealing with negative numbers. The remainder usually takes the sign of the dividend, while the modulo operation often takes the sign of the divisor. For positive numbers, they are generally the same.