Aliquot part

Aliquot part is a term used in number theory, a branch of mathematics, to describe a specific type of divisor of a number. An aliquot part of a number is a proper divisor of that number, meaning it divides the number evenly but is not equal to the number itself. The concept of aliquot parts is fundamental to the study of perfect numbers, amicable numbers, and sociable numbers.

Definition

An aliquot part of a number is any of the number's divisors excluding the number itself. For example, the aliquot parts of 12 are 1, 2, 3, 4, and 6. The term "aliquot" is derived from the Latin word aliquot, meaning "some, several". In mathematics, the term was first used by Euclid in his work Elements.

Properties

Aliquot parts have several interesting properties. For instance, if the sum of the aliquot parts of a number equals the number itself, that number is called a perfect number. An example of a perfect number is 6, whose aliquot parts (1, 2, and 3) sum to 6.

Two numbers are called amicable numbers if the sum of the aliquot parts of each number equals the other number. For example, 220 and 284 are amicable numbers because the sum of the aliquot parts of 220 is 284, and the sum of the aliquot parts of 284 is 220.

A set of numbers is called a sociable chain if the sum of the aliquot parts of each number in the set equals the next number in the set, and the last number in the set equals the first. For example, the numbers 12496, 14288, 15472, 14536, and 14264 form a sociable chain.

Applications

The concept of aliquot parts is used in the study of number theory, particularly in the investigation of perfect, amicable, and sociable numbers. It is also used in the Euclidean algorithm, a method for finding the greatest common divisor of two numbers.

See also

• Divisor
• Perfect number
• Amicable number
• Sociable number
• Euclidean algorithm