What is a divisor of a number ? We say that is a divisor of , if there exists an integer such that .
In general, the divisors of a number refer to the positive divisors, unless otherwise noted. Since the negative divisors will be the negative of a positive divisor (and vice versa), we shall just consider positive divisors.
We also tend to ignore 0 the possibility for any of these numbers to be 0. Since our definition above gives us that every integer is a divisor of .
Let the integer have a prime factorization , where are distinct prime numbers, and are positive integers. Let be a divisor of ; then any prime factor that divides must divide , hence it must be one of .
Without loss of generality, set . Let the highest power of that divides be . Then, divides , which in turn divides , hence, must divide , which means that . Thus, by considering all the prime factors of , we get that it must have the form where for all .
Conversely, given a number that has the form where for all it is clear that is a divisor of . As such, we have a complete classification of all the divisors.
How many divisors does the number have? From the above classification, we can set up a direct bijection between and sets of integers that satisfy . For each , there are possibilities. Hence, by the product rule, there are going to be divisors in all. The number of divisors of an integer is often denoted as the or .
How many divisors does the number have?
We have . Hence, from the above discussion, it has divisors.
We can list them out as 1, 2, 4, 8, 16, 5, 10, 20, 40, 80, 25, 50, 100, 200, 400, 125, 250, 500, 1000, 2000.
(Can you see why we listed out the divisors this way, instead of in increasing order?)
What is the sum of all divisors of the number ?
Consider the product when expanded out. From the classification of the divisors, each divisor would appear exactly once as a term. Moreover, every term would be a divisor of the number . Hence the product represents the sum of all the divisors of the number , which is .
(Pop quiz: How would you generalize this to find the sum of all divisors of the number ? It is sometimes denoted as or .)
Show that an integer has an odd number of divisors if and only if it is a perfect square.
Since , this product is odd if and only if every term is odd, which happens if and only if every value is even, which happens if and only if is a perfect square.
What is the smallest integer that has exactly 14 divisors?
Since , an integer has exactly 14 divisors if it has the form or . The smallest number in the first case and second case are, and , respectively. Hence 192 is the smallest integer that has exactly 14 divisors.