What are the divisors of 5408?

1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 169, 208, 338, 416, 676, 1352, 2704, 5408

There is a total of 18 positive divisors.
The sum of these divisors is 11529.
The arithmetic mean is 640.5.

15 even divisors

2, 4, 8, 16, 26, 32, 52, 104, 208, 338, 416, 676, 1352, 2704, 5408

3 odd divisors

1, 13, 169

How to compute the divisors of 5408?

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 5408 by each of the numbers from 1 to 5408 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)

5408 / 1 = 5408 (the remainder is 0, so 1 is a divisor of 5408)
5408 / 2 = 2704 (the remainder is 0, so 2 is a divisor of 5408)
5408 / 3 = 1802.6666666667 (the remainder is 2, so 3 is not a divisor of 5408)
...
5408 / 5407 = 1.0001849454411 (the remainder is 1, so 5407 is not a divisor of 5408)
5408 / 5408 = 1 (the remainder is 0, so 5408 is a divisor of 5408)

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 5408 (i.e. 73.539105243401). 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 $\frac{N}{d} = D$ )

5408 / 1 = 5408 (the remainder is 0, so 1 and 5408 are divisors of 5408)
5408 / 2 = 2704 (the remainder is 0, so 2 and 2704 are divisors of 5408)
5408 / 3 = 1802.6666666667 (the remainder is 2, so 3 is not a divisor of 5408)
...
5408 / 72 = 75.111111111111 (the remainder is 8, so 72 is not a divisor of 5408)
5408 / 73 = 74.082191780822 (the remainder is 6, so 73 is not a divisor of 5408)