A proper divisor is a number that when divided into another number leaves no remainder. Such a definition excludes all numbers greater than from being among its proper divisors, as they would always give a fractional result. Thus the proper divisors of 15, for example, are 1, 3, 5, and 15, although the number itself is usually omitted from the list, as is the number 1. So the proper divisors of 15 would generally be considered to be 3 and 5, both of which satisfy the inequality .
The reason why the number itself is excluded from the list of its proper divisors can be traced back to the Ancient Greek mathematicians, who conceptualised numbers as segments rather than as abstractions in their own right. Therefore, as the whole cannot be a part of itself, it cannot be a proper divisor. The Greeks would, however, have considered 1 to be included among the proper divisors.
Negative numbers can also be included in the list of proper divisors for a given integer, which for our example of 15 would be –3 and –5. So a complete list of proper divisors for 15 would be –5, –3, 3, and 5.