What are the divisors of 1586?

1, 2, 13, 26, 61, 122, 793, 1586

There is a total of 8 positive divisors.
The sum of these divisors is 2604.
The arithmetic mean is 325.5.

4 even divisors

2, 26, 122, 1586

4 odd divisors

1, 13, 61, 793

How to compute the divisors of 1586?

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 1586 by each of the numbers from 1 to 1586 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)

1586 / 1 = 1586 (the remainder is 0, so 1 is a divisor of 1586)
1586 / 2 = 793 (the remainder is 0, so 2 is a divisor of 1586)
1586 / 3 = 528.66666666667 (the remainder is 2, so 3 is not a divisor of 1586)
...
1586 / 1585 = 1.0006309148265 (the remainder is 1, so 1585 is not a divisor of 1586)
1586 / 1586 = 1 (the remainder is 0, so 1586 is a divisor of 1586)

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 1586 (i.e. 39.82461550348). 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 $\frac{N}{d} = D$ )

1586 / 1 = 1586 (the remainder is 0, so 1 and 1586 are divisors of 1586)
1586 / 2 = 793 (the remainder is 0, so 2 and 793 are divisors of 1586)
1586 / 3 = 528.66666666667 (the remainder is 2, so 3 is not a divisor of 1586)
...
1586 / 38 = 41.736842105263 (the remainder is 28, so 38 is not a divisor of 1586)
1586 / 39 = 40.666666666667 (the remainder is 26, so 39 is not a divisor of 1586)