What are the divisors of 1027?

1, 13, 79, 1027

There is a total of 4 positive divisors.
The sum of these divisors is 1120.
The arithmetic mean is 280.

4 odd divisors

1, 13, 79, 1027

How to compute the divisors of 1027?

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 1027 by each of the numbers from 1 to 1027 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)

1027 / 1 = 1027 (the remainder is 0, so 1 is a divisor of 1027)
1027 / 2 = 513.5 (the remainder is 1, so 2 is not a divisor of 1027)
1027 / 3 = 342.33333333333 (the remainder is 1, so 3 is not a divisor of 1027)
...
1027 / 1026 = 1.0009746588694 (the remainder is 1, so 1026 is not a divisor of 1027)
1027 / 1027 = 1 (the remainder is 0, so 1027 is a divisor of 1027)

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 1027 (i.e. 32.046840717924). 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 $\frac{N}{d} = D$ )

1027 / 1 = 1027 (the remainder is 0, so 1 and 1027 are divisors of 1027)
1027 / 2 = 513.5 (the remainder is 1, so 2 is not a divisor of 1027)
1027 / 3 = 342.33333333333 (the remainder is 1, so 3 is not a divisor of 1027)
...
1027 / 31 = 33.129032258065 (the remainder is 4, so 31 is not a divisor of 1027)
1027 / 32 = 32.09375 (the remainder is 3, so 32 is not a divisor of 1027)