Number Sense In division, why should the remainder not be greater than the divisor?

In division, the remainder represents what is left over after dividing a number (the dividend) by another number (the divisor). The fundamental reason why the remainder should not be greater than the divisor is based on the definition of division itself. When you divide a number \( a \) (the dividend) by another number \( b \) (the divisor), you are essentially determining how many times \( b \) can fit into \( a \) completely, and what is left over after that. The division can be expressed in the following equation: \[ a = b \times q + r \] where: - \( a \) is the dividend, - \( b \) is the divisor, - \( q \) is the quotient (the number of times \( b \) fits into \( a \)), - \( r \) is the remainder. For the equation to hold true, the remainder \( r \) must satisfy the condition: \[ 0 \leq r < b \] This means that the remainder must be non-negative and less than the divisor. If the remainder were equal to or greater than the divisor, it would imply that you could still divide further, which contradicts the definition of the remainder. For example, if you were dividing 10 by 3: - 3 fits into 10 three times (3 × 3 = 9), leaving a remainder of 1 (10 - 9 = 1). - The remainder (1) is less than the divisor (3). If the remainder were 3 or more, it would mean that you could still divide by 3, which indicates that the division process was not complete. Thus, the remainder must always be less than the divisor to ensure that the division is fully resolved.