Division can answer two questions. If 15 counters are shared between two people, how many whole counters does each person get? If the same counters are packed in pairs, how many pairs can you make? Both interpretations give a quotient of 7 and a remainder of 1. In the first, 2 counts the recipients. In the second, it counts the items in each group.
An integer quotient and an exact quotient describe different results
For a nonnegative integer dividend N and a positive integer divisor d, the quotient q is the largest whole number for which the product dq does not exceed N. The remainder r is what is left after subtracting that product. If r were at least d, another group would fit.
An exact quotient may be fractional. In arithmetic with rational numbers, 15 divided by 2 is 7.5. When counters must stay whole, record the quotient and the remainder separately. With a zero remainder, both interpretations give the same whole-number quotient.
Long division by hand with a remainder
Divide 157 by 6. Start with 15 tens, since 6 does not fit into the first digit alone. Two groups of six tens use 12 tens and leave 3 tens. Bring down the 7 units to make 37. Six groups of six use 36 and leave 1.
The 2 in the quotient stands for two tens. The 6 stands for six units. Multiplying 26 by 6 and adding 1 reconstructs 157. This check also works for exact divisions, where the remainder is zero.
What happens with signs or zero?
For two nonzero real numbers, like signs give a positive quotient and unlike signs give a negative one. Think of the quotient as the number that must multiply the divisor to recover the dividend. For example, dividing −24 by 6 gives −4 because 6 times −4 equals −24.
Zero divided by any nonzero number is zero. A nonzero number divided by zero has no possible quotient, since every number multiplied by zero gives zero. For zero divided by zero, that same test admits every number, so it does not define a unique quotient.
Which arithmetic properties survive division?
Dividing a number by 1 leaves it unchanged. Changing the order usually changes the result, as 12 divided by 3 gives 4 and 3 divided by 12 gives one quarter. Grouping matters too. Dividing 24 by 6 and then by 2 gives 2. Dividing 24 by the result of 6 divided by 2 gives 8. Integers are not closed under exact division because the result may be fractional.
