If someone has twelve marbles, these can be shared equally among three people. This is possible because #\blue 3# is a divisor of #\orange{12}#. The result of the division is called the quotient. In this case, the quotient is #\green 4#.
The number #\blue 3# is a divisor of #\orange{\orange{12}}#,
since #\orange{\orange{12}} \div \blue{3}# is exactly equal to #\green 4#.
#\blue 1# is also a divisor of #\orange{12}#,
since #\orange{12} \div \blue 1=\green{12}# is an integer.
The biggest divisor of #\orange{12}# is #\blue {12}# itself,
since #\orange{12} \div \blue{12}=\green 1#.
The divisors of #\orange{12}# are #\blue 1#, #\blue 2#, #\blue 3#, #\blue 4#, #\blue 6# and #\blue{12}#. \[\begin{array}{rclcccl}
\orange{12} \div \blue{1} &=& \green{12} &\text{ and }& \orange{12} \div \blue{\blue{12}} &=& \green 1 \\
\orange{12} \div \blue{3} &=& \green 4 &\text{and} &\orange{12} \div \blue{4} &=& \green 3 \\
\orange{12} \div \blue{6} &=& \green 2 & \text{ and }& \orange{12} \div \blue{2} &=&\green 6 \end{array}\]
An integer #\blue{a}# is a divisor of an integer #\orange{b}# if #\orange{b} \div \blue{a}# results in an integer #\green q#, or in other words #\blue{a} \times \green{q} = \orange{b}#.
The result of the division #\green{q}# is known as the quotient.
If we divide a number #\orange{b}# by a number #\blue{a}# but #\blue{a}# is not a divisor of #\orange{b}#, the result is a quotient #\green q# and a remainder #\purple{r}#, such that
\[ \orange{b} = \blue{a} \times \green{q} + \purple{r}\]
The remainder #\purple r# is a non-negative number smaller than #\blue{a}#.
To find the quotient and remainder when dividing two numbers #\orange{b}# and #\blue{a}# where #\blue{a}# is not a divisor of #\orange{b}#, find the largest number #\green{q}# such that #\blue{a} \times \green{q}# is not larger than #\orange{b}#. Then the remainder #\purple{r} = \orange{b} - \blue{a} \times \green{q}#.
Using a calculator, the quotient #\green q# is found by calculating #\orange{b} \div \blue{a}# and rounding the answer down, so #\green q# is the number without the decimals. Then, the remainder #\purple r# is calculated as #\orange{b} - \blue{a} \times \green{q}#.
Example
The quotient of #\orange{38}# and #\blue{7}# is #\green 5# and the remainder is #\purple 3#.
\[\orange{38} = \blue{7} \times \green{5} + \purple{3}\]
With a calculator we find #\orange{38} \div \blue{7} \approx 5.428571428571429# hence the quotient is #\green 5# which leaves the remainder # \orange{38} -\blue{7} \times \green{5} = \purple{3}#.
Is #15# a divisor of #15#?
Yes
Since #15 \div 15=1#, we see that #15# is a divisor of #15# and #1# is the quotient.
Divisors
