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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 3-4, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.

This theorem is applicable to integer numbers.

Theorem

For any two integers D and d, with d non-zero, there exist two uniquely determined integers q and r, with r<d, such that:

equation

where:

Ddividendinteger we divide
ddivisorinteger we divide by
qquotientnumber of integer times d goes evenly into D
rremainderleft-over from D that is too small for d to fit in one more time

From the definition of the remainder we see why we had to specify that r<d in the theorem. If this condition is not fulfilled then q may simply be increased. However, if it is fulfilled, then the integers q and r are uniquely determined and the theorem is true in every detail. Notice how the condition is logically connected to the uniqueness.

Example

Take the numbers D = 40 and d = 6.

Then, there exist q = 8 and r = 2 (and notice they are unique if we ask for r<d), such that:

equation

This theorem does not apply to numbers that are not integer, since for those numbers we can always use long division (in which the concept of a remainder does not exist).

Definitions pertaining to integer division

A number n is divisible by a number m if the remainder of the integer division of n by m is 0.

Example: 6 is divisible by 3.

A number a is a multiple of a number m if it is divisible by it.

Example: 6 is a multiple of 3.

Fact: A non-zero integer has an infinity of multiples.

Fact: We obtain a list of multiples of a number by 'counting by it'.

Example: 4, 8, 12, 16, 20, ... are positive multiples of 4.

Fact: Zero is a multiple of any integer because:

equation

Students often confuse this notion with the fact that division by zero is undefined. The consequence of this fact, however, is that zero is not a divisor of any number. If we understand correctly the definitions of 'multiple' and 'divisor' there is nothing confusing anymore.

A number a is a divisor (factor) of a number b if b is a multiple of a.

Example: 3 is a divisor (factor) of 6.

Fact: Any integer has a finite number of divisors (factors).

A prime divisor (factor) of an integer is a divisor (factor) that is prime.

A proper divisor (factor) of an integer is a divisor (factor) that is different from the number itself.

Example: 4 is a proper divisor of 12 but 12 is not a proper divisor of 12.

Fact: Since any number multiplied by 0 gives a result of 0, it follows that 0 is a multiple of any integer.

Fact: Since there is no integer that, multiplied by another gives 0, there is no integer that has 0 for a divisor.