Divisor - Further Notions and Facts

Further Notions and Facts

There are some elementary rules:

• If and, then . This is the transitive relation.
• If and, then or .
• If and, then it is NOT always true that (e.g. and but 5 does not divide 6). However, when and, then is true, as is .

If, and gcd, then . This is called Euclid's lemma.

If is a prime number and then or (or both).

A positive divisor of which is different from is called a proper divisor or an aliquot part of . A number that does not evenly divide but leaves a remainder is called an aliquant part of .

An integer whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.

Any positive divisor of is a product of prime divisors of raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than, and abundant if this sum exceeds .

The total number of positive divisors of is a multiplicative function, meaning that when two numbers and are relatively prime, then . For instance, ; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42). However the number of positive divisors is not a totally multiplicative function: if the two numbers and share a common divisor, then it might not be true that . The sum of the positive divisors of is another multiplicative function (e.g. ). Both of these functions are examples of divisor functions.

If the prime factorization of is given by

then the number of positive divisors of is

and each of the divisors has the form

where for each

It can be shown that for any natural the inequality holds.

Also it can be shown that

One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about .