What are the divisors of 100?

1, 2, 4, 5, 10, 20, 25, 50, 100

There is a total of 9 positive divisors.
The sum of these divisors is 217.
The arithmetic mean is 24.111111111111.

6 even divisors

2, 4, 10, 20, 50, 100

3 odd divisors

1, 5, 25

How to compute the divisors of 100?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 100 by each of the numbers from 1 to 100 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

100 / 1 = 100 (the remainder is 0, so 1 is a divisor of 100)
100 / 2 = 50 (the remainder is 0, so 2 is a divisor of 100)
100 / 3 = 33.333333333333 (the remainder is 1, so 3 is not a divisor of 100)
...
100 / 99 = 1.010101010101 (the remainder is 1, so 99 is not a divisor of 100)
100 / 100 = 1 (the remainder is 0, so 100 is a divisor of 100)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 100 (i.e. 10). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

100 / 1 = 100 (the remainder is 0, so 1 and 100 are divisors of 100)
100 / 2 = 50 (the remainder is 0, so 2 and 50 are divisors of 100)
100 / 3 = 33.333333333333 (the remainder is 1, so 3 is not a divisor of 100)
...
100 / 9 = 11.111111111111 (the remainder is 1, so 9 is not a divisor of 100)
100 / 10 = 10 (the remainder is 0, so 10 and 10 are divisors of 100)