What are the divisors of 1012?

1, 2, 4, 11, 22, 23, 44, 46, 92, 253, 506, 1012

There is a total of 12 positive divisors.
The sum of these divisors is 2016.
The arithmetic mean is 168.

8 even divisors

2, 4, 22, 44, 46, 92, 506, 1012

4 odd divisors

1, 11, 23, 253

How to compute the divisors of 1012?

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 1012 by each of the numbers from 1 to 1012 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)

1012 / 1 = 1012 (the remainder is 0, so 1 is a divisor of 1012)
1012 / 2 = 506 (the remainder is 0, so 2 is a divisor of 1012)
1012 / 3 = 337.33333333333 (the remainder is 1, so 3 is not a divisor of 1012)
...
1012 / 1011 = 1.0009891196835 (the remainder is 1, so 1011 is not a divisor of 1012)
1012 / 1012 = 1 (the remainder is 0, so 1012 is a divisor of 1012)

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 1012 (i.e. 31.811947441174). 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 $\frac{N}{d} = D$ )

1012 / 1 = 1012 (the remainder is 0, so 1 and 1012 are divisors of 1012)
1012 / 2 = 506 (the remainder is 0, so 2 and 506 are divisors of 1012)
1012 / 3 = 337.33333333333 (the remainder is 1, so 3 is not a divisor of 1012)
...
1012 / 30 = 33.733333333333 (the remainder is 22, so 30 is not a divisor of 1012)
1012 / 31 = 32.645161290323 (the remainder is 20, so 31 is not a divisor of 1012)