What are the divisors of 1359?

1, 3, 9, 151, 453, 1359

There is a total of 6 positive divisors.
The sum of these divisors is 1976.
The arithmetic mean is 329.33333333333.

6 odd divisors

1, 3, 9, 151, 453, 1359

How to compute the divisors of 1359?

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 1359 by each of the numbers from 1 to 1359 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)

1359 / 1 = 1359 (the remainder is 0, so 1 is a divisor of 1359)
1359 / 2 = 679.5 (the remainder is 1, so 2 is not a divisor of 1359)
1359 / 3 = 453 (the remainder is 0, so 3 is a divisor of 1359)
...
1359 / 1358 = 1.000736377025 (the remainder is 1, so 1358 is not a divisor of 1359)
1359 / 1359 = 1 (the remainder is 0, so 1359 is a divisor of 1359)

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 1359 (i.e. 36.864617182334). 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 $\frac{N}{d} = D$ )

1359 / 1 = 1359 (the remainder is 0, so 1 and 1359 are divisors of 1359)
1359 / 2 = 679.5 (the remainder is 1, so 2 is not a divisor of 1359)
1359 / 3 = 453 (the remainder is 0, so 3 and 453 are divisors of 1359)
...
1359 / 35 = 38.828571428571 (the remainder is 29, so 35 is not a divisor of 1359)
1359 / 36 = 37.75 (the remainder is 27, so 36 is not a divisor of 1359)