What are the divisors of 921?

1, 3, 307, 921

There is a total of 4 positive divisors.
The sum of these divisors is 1232.
The arithmetic mean is 308.

4 odd divisors

1, 3, 307, 921

How to compute the divisors of 921?

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 921 by each of the numbers from 1 to 921 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)

921 / 1 = 921 (the remainder is 0, so 1 is a divisor of 921)
921 / 2 = 460.5 (the remainder is 1, so 2 is not a divisor of 921)
921 / 3 = 307 (the remainder is 0, so 3 is a divisor of 921)
...
921 / 920 = 1.0010869565217 (the remainder is 1, so 920 is not a divisor of 921)
921 / 921 = 1 (the remainder is 0, so 921 is a divisor of 921)

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 921 (i.e. 30.347981810987). 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 $\frac{N}{d} = D$ )

921 / 1 = 921 (the remainder is 0, so 1 and 921 are divisors of 921)
921 / 2 = 460.5 (the remainder is 1, so 2 is not a divisor of 921)
921 / 3 = 307 (the remainder is 0, so 3 and 307 are divisors of 921)
...
921 / 29 = 31.758620689655 (the remainder is 22, so 29 is not a divisor of 921)
921 / 30 = 30.7 (the remainder is 21, so 30 is not a divisor of 921)