What are the divisors of 1619?

1, 1619

There is a total of 2 positive divisors.
The sum of these divisors is 1620.
The arithmetic mean is 810.

2 odd divisors

1, 1619

How to compute the divisors of 1619?

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 1619 by each of the numbers from 1 to 1619 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)

1619 / 1 = 1619 (the remainder is 0, so 1 is a divisor of 1619)
1619 / 2 = 809.5 (the remainder is 1, so 2 is not a divisor of 1619)
1619 / 3 = 539.66666666667 (the remainder is 2, so 3 is not a divisor of 1619)
...
1619 / 1618 = 1.0006180469716 (the remainder is 1, so 1618 is not a divisor of 1619)
1619 / 1619 = 1 (the remainder is 0, so 1619 is a divisor of 1619)

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 1619 (i.e. 40.236799077461). 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 $\frac{N}{d} = D$ )

1619 / 1 = 1619 (the remainder is 0, so 1 and 1619 are divisors of 1619)
1619 / 2 = 809.5 (the remainder is 1, so 2 is not a divisor of 1619)
1619 / 3 = 539.66666666667 (the remainder is 2, so 3 is not a divisor of 1619)
...
1619 / 39 = 41.512820512821 (the remainder is 20, so 39 is not a divisor of 1619)
1619 / 40 = 40.475 (the remainder is 19, so 40 is not a divisor of 1619)