What are the divisors of 1531?

1, 1531

There is a total of 2 positive divisors.
The sum of these divisors is 1532.
The arithmetic mean is 766.

2 odd divisors

1, 1531

How to compute the divisors of 1531?

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 1531 by each of the numbers from 1 to 1531 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)

1531 / 1 = 1531 (the remainder is 0, so 1 is a divisor of 1531)
1531 / 2 = 765.5 (the remainder is 1, so 2 is not a divisor of 1531)
1531 / 3 = 510.33333333333 (the remainder is 1, so 3 is not a divisor of 1531)
...
1531 / 1530 = 1.0006535947712 (the remainder is 1, so 1530 is not a divisor of 1531)
1531 / 1531 = 1 (the remainder is 0, so 1531 is a divisor of 1531)

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 1531 (i.e. 39.127995093028). 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 $\frac{N}{d} = D$ )

1531 / 1 = 1531 (the remainder is 0, so 1 and 1531 are divisors of 1531)
1531 / 2 = 765.5 (the remainder is 1, so 2 is not a divisor of 1531)
1531 / 3 = 510.33333333333 (the remainder is 1, so 3 is not a divisor of 1531)
...
1531 / 38 = 40.289473684211 (the remainder is 11, so 38 is not a divisor of 1531)
1531 / 39 = 39.25641025641 (the remainder is 10, so 39 is not a divisor of 1531)