What are the divisors of 1459?

1, 1459

There is a total of 2 positive divisors.
The sum of these divisors is 1460.
The arithmetic mean is 730.

2 odd divisors

1, 1459

How to compute the divisors of 1459?

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 1459 by each of the numbers from 1 to 1459 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)

1459 / 1 = 1459 (the remainder is 0, so 1 is a divisor of 1459)
1459 / 2 = 729.5 (the remainder is 1, so 2 is not a divisor of 1459)
1459 / 3 = 486.33333333333 (the remainder is 1, so 3 is not a divisor of 1459)
...
1459 / 1458 = 1.0006858710562 (the remainder is 1, so 1458 is not a divisor of 1459)
1459 / 1459 = 1 (the remainder is 0, so 1459 is a divisor of 1459)

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 1459 (i.e. 38.196858509569). 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 $\frac{N}{d} = D$ )

1459 / 1 = 1459 (the remainder is 0, so 1 and 1459 are divisors of 1459)
1459 / 2 = 729.5 (the remainder is 1, so 2 is not a divisor of 1459)
1459 / 3 = 486.33333333333 (the remainder is 1, so 3 is not a divisor of 1459)
...
1459 / 37 = 39.432432432432 (the remainder is 16, so 37 is not a divisor of 1459)
1459 / 38 = 38.394736842105 (the remainder is 15, so 38 is not a divisor of 1459)