What are the divisors of 1559?

1, 1559

There is a total of 2 positive divisors.
The sum of these divisors is 1560.
The arithmetic mean is 780.

2 odd divisors

1, 1559

How to compute the divisors of 1559?

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 1559 by each of the numbers from 1 to 1559 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)

1559 / 1 = 1559 (the remainder is 0, so 1 is a divisor of 1559)
1559 / 2 = 779.5 (the remainder is 1, so 2 is not a divisor of 1559)
1559 / 3 = 519.66666666667 (the remainder is 2, so 3 is not a divisor of 1559)
...
1559 / 1558 = 1.0006418485237 (the remainder is 1, so 1558 is not a divisor of 1559)
1559 / 1559 = 1 (the remainder is 0, so 1559 is a divisor of 1559)

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 1559 (i.e. 39.484174044799). 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 $\frac{N}{d} = D$ )

1559 / 1 = 1559 (the remainder is 0, so 1 and 1559 are divisors of 1559)
1559 / 2 = 779.5 (the remainder is 1, so 2 is not a divisor of 1559)
1559 / 3 = 519.66666666667 (the remainder is 2, so 3 is not a divisor of 1559)
...
1559 / 38 = 41.026315789474 (the remainder is 1, so 38 is not a divisor of 1559)
1559 / 39 = 39.974358974359 (the remainder is 38, so 39 is not a divisor of 1559)