What are the divisors of 200?

1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200

There is a total of 12 positive divisors.
The sum of these divisors is 465.
The arithmetic mean is 38.75.

9 even divisors

2, 4, 8, 10, 20, 40, 50, 100, 200

3 odd divisors

1, 5, 25

How to compute the divisors of 200?

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 200 by each of the numbers from 1 to 200 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)

200 / 1 = 200 (the remainder is 0, so 1 is a divisor of 200)
200 / 2 = 100 (the remainder is 0, so 2 is a divisor of 200)
200 / 3 = 66.666666666667 (the remainder is 2, so 3 is not a divisor of 200)
...
200 / 199 = 1.0050251256281 (the remainder is 1, so 199 is not a divisor of 200)
200 / 200 = 1 (the remainder is 0, so 200 is a divisor of 200)

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 200 (i.e. 14.142135623731). 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 $\frac{N}{d} = D$ )

200 / 1 = 200 (the remainder is 0, so 1 and 200 are divisors of 200)
200 / 2 = 100 (the remainder is 0, so 2 and 100 are divisors of 200)
200 / 3 = 66.666666666667 (the remainder is 2, so 3 is not a divisor of 200)
...
200 / 13 = 15.384615384615 (the remainder is 5, so 13 is not a divisor of 200)
200 / 14 = 14.285714285714 (the remainder is 4, so 14 is not a divisor of 200)