What are the divisors of 931?

1, 7, 19, 49, 133, 931

There is a total of 6 positive divisors.
The sum of these divisors is 1140.
The arithmetic mean is 190.

6 odd divisors

1, 7, 19, 49, 133, 931

How to compute the divisors of 931?

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 931 by each of the numbers from 1 to 931 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)

931 / 1 = 931 (the remainder is 0, so 1 is a divisor of 931)
931 / 2 = 465.5 (the remainder is 1, so 2 is not a divisor of 931)
931 / 3 = 310.33333333333 (the remainder is 1, so 3 is not a divisor of 931)
...
931 / 930 = 1.0010752688172 (the remainder is 1, so 930 is not a divisor of 931)
931 / 931 = 1 (the remainder is 0, so 931 is a divisor of 931)

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 931 (i.e. 30.512292604785). 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 $\frac{N}{d} = D$ )

931 / 1 = 931 (the remainder is 0, so 1 and 931 are divisors of 931)
931 / 2 = 465.5 (the remainder is 1, so 2 is not a divisor of 931)
931 / 3 = 310.33333333333 (the remainder is 1, so 3 is not a divisor of 931)
...
931 / 29 = 32.103448275862 (the remainder is 3, so 29 is not a divisor of 931)
931 / 30 = 31.033333333333 (the remainder is 1, so 30 is not a divisor of 931)