is defined as the number of positive divisors of , including and .
Properties
Theorem
Given the .
Proof:
Any divisor must be in the form with . So for each there are choices for . Since they are independent you multiply.
Theorem there are distinct ordered pairs such that .
Proof:
Given that are divisors of , they must have the form and since
There are options for each :
So you multiply number of choices together.
Theorem
Proof:Case 1
You can split the divisors into pairs so their product is multiplying all together you get Then multiply by yielding
Case 2
You can split the divisors into pairs so their product is multiplying all together you get .
Theorem
Proof:
Let be the divisors of . The rest of the divisors are . In each set there are numbers so (or if is a perfect square) Since are distinct positive integers there can be at most so .
Theorem is odd .
Proof:
Using we can deduce that therefore is square.
Proof:
Let and with . Meaning they share no prime factors. and therefore . Since they share no prime factors so there is no crossover in primes and this is the proper prime factorisation.
(If weren’t coprime they would share factors and this would not be the proper prime factorisation: you could get terms. This means that is not completely multiplicative).