What are the divisors of 1016?

1, 2, 4, 8, 127, 254, 508, 1016

There is a total of 8 positive divisors.
The sum of these divisors is 1920.
The arithmetic mean is 240.

6 even divisors

2, 4, 8, 254, 508, 1016

2 odd divisors

1, 127

How to compute the divisors of 1016?

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 1016 by each of the numbers from 1 to 1016 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)

1016 / 1 = 1016 (the remainder is 0, so 1 is a divisor of 1016)
1016 / 2 = 508 (the remainder is 0, so 2 is a divisor of 1016)
1016 / 3 = 338.66666666667 (the remainder is 2, so 3 is not a divisor of 1016)
...
1016 / 1015 = 1.0009852216749 (the remainder is 1, so 1015 is not a divisor of 1016)
1016 / 1016 = 1 (the remainder is 0, so 1016 is a divisor of 1016)

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 1016 (i.e. 31.874754901018). 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 $\frac{N}{d} = D$ )

1016 / 1 = 1016 (the remainder is 0, so 1 and 1016 are divisors of 1016)
1016 / 2 = 508 (the remainder is 0, so 2 and 508 are divisors of 1016)
1016 / 3 = 338.66666666667 (the remainder is 2, so 3 is not a divisor of 1016)
...
1016 / 30 = 33.866666666667 (the remainder is 26, so 30 is not a divisor of 1016)
1016 / 31 = 32.774193548387 (the remainder is 24, so 31 is not a divisor of 1016)