theory of divisors

mathematics

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June 25, 2026

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In mathematics, a divisor is a number that divides another number evenly, leaving no remainder. For example, 3 and 6 are divisors of 12. Numbers can be classified based on the sum of their proper divisors (divisors excluding the number itself).12

Abundant numbers have a sum of proper divisors greater than the number itself (e.g., 24, whose divisors sum to 36).3
Deficient numbers have a sum of proper divisors less than the number itself (e.g., 8, whose divisors sum to 7; or 32, whose divisors sum to 31).43
Perfect numbers are integers that equal the sum of their proper divisors (e.g., 6, where 1 + 2 + 3 = 6; or 28, where 1 + 2 + 4 + 7 + 14 = 28).3

The study of integers and their properties, including divisors, is the focus of number theory.5

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$A page from a first-grade workbook typical of “new math” might state: “Draw connecting lines from triangles in the first set to triangles in the second set. Are the two sets equivalent in number?”$
arithmetic
Theory of divisors
At this point an interesting development occurs, for, so long as only additions and multiplications are performed with integers, the resulting numbers are invariably themselves integers—that is, numbers of the same kind as their antecedents. This characteristic changes drastically, however, as soon…
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At this point an interesting development occurs, for, so long as only additions and multiplications are performed with integers, the resulting numbers are invariably themselves integers—that is, numbers of the same kind as their antecedents. This characteristic changes drastically, however, as soon…
Read More