Division - Properties of Division

The terms of division consist of the Dividend (the total number to be divided), the Divisor (the number of groups or the number in each group), the Quotient (the answer), and the Remainder (the amount left over). For example, in $17 \div 5 = 3$ with a remainder of $2$, $17$ is the dividend and $5$ is the divisor. Visually, imagine $17$ dots arranged in rows of $5$; you will have $3$ full rows and $2$ extra dots leftover.

Property of 1: When any number is divided by $1$, the quotient is the number itself. For example, $45 \div 1 = 45$. This means if you have $45$ items and put them into $1$ big group, that group still contains all $45$ items.

Property of Self: When a non-zero number is divided by itself, the quotient is always $1$. For example, $12 \div 12 = 1$. Visually, if you have $12$ chocolates and share them among $12$ friends, each friend gets exactly $1$ chocolate.

Division of Zero: When zero is divided by any number (except zero), the quotient is always $0$. For example, $0 \div 8 = 0$. If you have $0$ toys to distribute among $8$ children, each child will receive $0$ toys.

Division by Zero: Dividing a number by zero is not defined and is not possible in mathematics. You cannot take a quantity and split it into zero groups or share it with zero people. In a division bracket, you can never have $0$ as the divisor on the outside.

Inverse Relation: Division is the inverse operation of multiplication. Every multiplication fact provides a division fact. If $8 \times 4 = 32$, then it follows that $32 \div 4 = 8$ and $32 \div 8 = 4$. This can be visualized as an array of dots where the total ($32$) can be split into either rows or columns.

The Remainder Rule: In any division problem, the Remainder must always be smaller than the Divisor ($Remainder < Divisor$). If the remainder is equal to or larger than the divisor, it means another full group could have been made. For example, when dividing by $4$, the only possible remainders are $0, 1, 2,$ or $3$.