What are the divisors of 771?

1, 3, 257, 771

There is a total of 4 positive divisors.
The sum of these divisors is 1032.
The arithmetic mean is 258.

4 odd divisors

1, 3, 257, 771

How to compute the divisors of 771?

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 771 by each of the numbers from 1 to 771 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)

771 / 1 = 771 (the remainder is 0, so 1 is a divisor of 771)
771 / 2 = 385.5 (the remainder is 1, so 2 is not a divisor of 771)
771 / 3 = 257 (the remainder is 0, so 3 is a divisor of 771)
...
771 / 770 = 1.0012987012987 (the remainder is 1, so 770 is not a divisor of 771)
771 / 771 = 1 (the remainder is 0, so 771 is a divisor of 771)

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 771 (i.e. 27.76688675383). 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 $\frac{N}{d} = D$ )

771 / 1 = 771 (the remainder is 0, so 1 and 771 are divisors of 771)
771 / 2 = 385.5 (the remainder is 1, so 2 is not a divisor of 771)
771 / 3 = 257 (the remainder is 0, so 3 and 257 are divisors of 771)
...
771 / 26 = 29.653846153846 (the remainder is 17, so 26 is not a divisor of 771)
771 / 27 = 28.555555555556 (the remainder is 15, so 27 is not a divisor of 771)