What are the divisors of 1019?

1, 1019

There is a total of 2 positive divisors.
The sum of these divisors is 1020.
The arithmetic mean is 510.

2 odd divisors

1, 1019

How to compute the divisors of 1019?

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 1019 by each of the numbers from 1 to 1019 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)

1019 / 1 = 1019 (the remainder is 0, so 1 is a divisor of 1019)
1019 / 2 = 509.5 (the remainder is 1, so 2 is not a divisor of 1019)
1019 / 3 = 339.66666666667 (the remainder is 2, so 3 is not a divisor of 1019)
...
1019 / 1018 = 1.0009823182711 (the remainder is 1, so 1018 is not a divisor of 1019)
1019 / 1019 = 1 (the remainder is 0, so 1019 is a divisor of 1019)

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 1019 (i.e. 31.921779399025). 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 $\frac{N}{d} = D$ )

1019 / 1 = 1019 (the remainder is 0, so 1 and 1019 are divisors of 1019)
1019 / 2 = 509.5 (the remainder is 1, so 2 is not a divisor of 1019)
1019 / 3 = 339.66666666667 (the remainder is 2, so 3 is not a divisor of 1019)
...
1019 / 30 = 33.966666666667 (the remainder is 29, so 30 is not a divisor of 1019)
1019 / 31 = 32.870967741935 (the remainder is 27, so 31 is not a divisor of 1019)