What are the divisors of 1600?

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800, 1600

There is a total of 21 positive divisors.
The sum of these divisors is 3937.
The arithmetic mean is 187.47619047619.

18 even divisors

2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800, 1600

3 odd divisors

1, 5, 25

How to compute the divisors of 1600?

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 1600 by each of the numbers from 1 to 1600 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)

1600 / 1 = 1600 (the remainder is 0, so 1 is a divisor of 1600)
1600 / 2 = 800 (the remainder is 0, so 2 is a divisor of 1600)
1600 / 3 = 533.33333333333 (the remainder is 1, so 3 is not a divisor of 1600)
...
1600 / 1599 = 1.0006253908693 (the remainder is 1, so 1599 is not a divisor of 1600)
1600 / 1600 = 1 (the remainder is 0, so 1600 is a divisor of 1600)

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 1600 (i.e. 40). 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 $\frac{N}{d} = D$ )

1600 / 1 = 1600 (the remainder is 0, so 1 and 1600 are divisors of 1600)
1600 / 2 = 800 (the remainder is 0, so 2 and 800 are divisors of 1600)
1600 / 3 = 533.33333333333 (the remainder is 1, so 3 is not a divisor of 1600)
...
1600 / 39 = 41.025641025641 (the remainder is 1, so 39 is not a divisor of 1600)
1600 / 40 = 40 (the remainder is 0, so 40 and 40 are divisors of 1600)