Division of whole numbers can be done in two ways. (Long division combines the two.)
keep subtracting the divisor
Keep subtracting the divisor until what is left is smaller than the divisor. What is left over is the remainder. The quotient is the number of times you subtracted.
For example, to find
divided by
, do
,
,
,
. Stop now because
is less than
. There are four subtractions so the quotient is
. The remainder is
.
Another example: to divide
by
, do
,
,
,
. Stop because
is less than
. There are four subtractions so the quotient is
. The remainder is
.
Now for division by zero. When you subtract zero from a whole number larger than zero, you get the same whole number back. So you can subtract again and again without making any progress.
For example, to find
divided by
, do
,
,
, …, forever. The result is never smaller than
no matter how many times you subtract.
subtract the largest possible multiple of the divisor
To divide two whole numbers both greater than zero, find the largest multiple of the divisor that is not larger than the number being divided. That multiple is the quotient. The remainder is the difference between the number being divided and the product of the quotient and divisor.
For example, divide
by
. The multiples of
are
,
,
,
,
,
,
, …. The largest multiple less than
is
. The quotient is 4 because
. The remainder is
. So
divided by
is the quotient
with remainder
.
Another example: to divide
by
, do
, so the quotient is
and the remainder is zero.
When dividing by zero, all the multiples of zero are zero, so there is no largest one less than the number to be divided.
For example, to divide
by
, all of these multiples are zero and are less than
:
,
,
,
, …,
, …,
, …. The quotient cannot be the largest of
,
,
,
, …,
, …,
, …, because there is no largest.
So do not divide by zero!
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