What are the divisors of 1597?

1, 1597

There is a total of 2 positive divisors.
The sum of these divisors is 1598.
The arithmetic mean is 799.

2 odd divisors

1, 1597

How to compute the divisors of 1597?

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 1597 by each of the numbers from 1 to 1597 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)

1597 / 1 = 1597 (the remainder is 0, so 1 is a divisor of 1597)
1597 / 2 = 798.5 (the remainder is 1, so 2 is not a divisor of 1597)
1597 / 3 = 532.33333333333 (the remainder is 1, so 3 is not a divisor of 1597)
...
1597 / 1596 = 1.000626566416 (the remainder is 1, so 1596 is not a divisor of 1597)
1597 / 1597 = 1 (the remainder is 0, so 1597 is a divisor of 1597)

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 1597 (i.e. 39.962482405376). 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 $\frac{N}{d} = D$ )

1597 / 1 = 1597 (the remainder is 0, so 1 and 1597 are divisors of 1597)
1597 / 2 = 798.5 (the remainder is 1, so 2 is not a divisor of 1597)
1597 / 3 = 532.33333333333 (the remainder is 1, so 3 is not a divisor of 1597)
...
1597 / 38 = 42.026315789474 (the remainder is 1, so 38 is not a divisor of 1597)
1597 / 39 = 40.948717948718 (the remainder is 37, so 39 is not a divisor of 1597)