What are the divisors of 1021?

1, 1021

There is a total of 2 positive divisors.
The sum of these divisors is 1022.
The arithmetic mean is 511.

2 odd divisors

1, 1021

How to compute the divisors of 1021?

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 1021 by each of the numbers from 1 to 1021 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)

1021 / 1 = 1021 (the remainder is 0, so 1 is a divisor of 1021)
1021 / 2 = 510.5 (the remainder is 1, so 2 is not a divisor of 1021)
1021 / 3 = 340.33333333333 (the remainder is 1, so 3 is not a divisor of 1021)
...
1021 / 1020 = 1.0009803921569 (the remainder is 1, so 1020 is not a divisor of 1021)
1021 / 1021 = 1 (the remainder is 0, so 1021 is a divisor of 1021)

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 1021 (i.e. 31.953090617341). 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 $\frac{N}{d} = D$ )

1021 / 1 = 1021 (the remainder is 0, so 1 and 1021 are divisors of 1021)
1021 / 2 = 510.5 (the remainder is 1, so 2 is not a divisor of 1021)
1021 / 3 = 340.33333333333 (the remainder is 1, so 3 is not a divisor of 1021)
...
1021 / 30 = 34.033333333333 (the remainder is 1, so 30 is not a divisor of 1021)
1021 / 31 = 32.935483870968 (the remainder is 29, so 31 is not a divisor of 1021)