What are the divisors of 1077?

1, 3, 359, 1077

There is a total of 4 positive divisors.
The sum of these divisors is 1440.
The arithmetic mean is 360.

4 odd divisors

1, 3, 359, 1077

How to compute the divisors of 1077?

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 1077 by each of the numbers from 1 to 1077 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)

1077 / 1 = 1077 (the remainder is 0, so 1 is a divisor of 1077)
1077 / 2 = 538.5 (the remainder is 1, so 2 is not a divisor of 1077)
1077 / 3 = 359 (the remainder is 0, so 3 is a divisor of 1077)
...
1077 / 1076 = 1.0009293680297 (the remainder is 1, so 1076 is not a divisor of 1077)
1077 / 1077 = 1 (the remainder is 0, so 1077 is a divisor of 1077)

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 1077 (i.e. 32.817678162844). 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 $\frac{N}{d} = D$ )

1077 / 1 = 1077 (the remainder is 0, so 1 and 1077 are divisors of 1077)
1077 / 2 = 538.5 (the remainder is 1, so 2 is not a divisor of 1077)
1077 / 3 = 359 (the remainder is 0, so 3 and 359 are divisors of 1077)
...
1077 / 31 = 34.741935483871 (the remainder is 23, so 31 is not a divisor of 1077)
1077 / 32 = 33.65625 (the remainder is 21, so 32 is not a divisor of 1077)