What are the divisors of 5205?

1, 3, 5, 15, 347, 1041, 1735, 5205

There is a total of 8 positive divisors.
The sum of these divisors is 8352.
The arithmetic mean is 1044.

8 odd divisors

1, 3, 5, 15, 347, 1041, 1735, 5205

How to compute the divisors of 5205?

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 5205 by each of the numbers from 1 to 5205 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)

5205 / 1 = 5205 (the remainder is 0, so 1 is a divisor of 5205)
5205 / 2 = 2602.5 (the remainder is 1, so 2 is not a divisor of 5205)
5205 / 3 = 1735 (the remainder is 0, so 3 is a divisor of 5205)
...
5205 / 5204 = 1.000192159877 (the remainder is 1, so 5204 is not a divisor of 5205)
5205 / 5205 = 1 (the remainder is 0, so 5205 is a divisor of 5205)

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 5205 (i.e. 72.145685941711). 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 $\frac{N}{d} = D$ )

5205 / 1 = 5205 (the remainder is 0, so 1 and 5205 are divisors of 5205)
5205 / 2 = 2602.5 (the remainder is 1, so 2 is not a divisor of 5205)
5205 / 3 = 1735 (the remainder is 0, so 3 and 1735 are divisors of 5205)
...
5205 / 71 = 73.30985915493 (the remainder is 22, so 71 is not a divisor of 5205)
5205 / 72 = 72.291666666667 (the remainder is 21, so 72 is not a divisor of 5205)