What are the divisors of 1456?

1, 2, 4, 7, 8, 13, 14, 16, 26, 28, 52, 56, 91, 104, 112, 182, 208, 364, 728, 1456

There is a total of 20 positive divisors.
The sum of these divisors is 3472.
The arithmetic mean is 173.6.

16 even divisors

2, 4, 8, 14, 16, 26, 28, 52, 56, 104, 112, 182, 208, 364, 728, 1456

4 odd divisors

1, 7, 13, 91

How to compute the divisors of 1456?

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 1456 by each of the numbers from 1 to 1456 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)

1456 / 1 = 1456 (the remainder is 0, so 1 is a divisor of 1456)
1456 / 2 = 728 (the remainder is 0, so 2 is a divisor of 1456)
1456 / 3 = 485.33333333333 (the remainder is 1, so 3 is not a divisor of 1456)
...
1456 / 1455 = 1.0006872852234 (the remainder is 1, so 1455 is not a divisor of 1456)
1456 / 1456 = 1 (the remainder is 0, so 1456 is a divisor of 1456)

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 1456 (i.e. 38.157568056678). 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 $\frac{N}{d} = D$ )

1456 / 1 = 1456 (the remainder is 0, so 1 and 1456 are divisors of 1456)
1456 / 2 = 728 (the remainder is 0, so 2 and 728 are divisors of 1456)
1456 / 3 = 485.33333333333 (the remainder is 1, so 3 is not a divisor of 1456)
...
1456 / 37 = 39.351351351351 (the remainder is 13, so 37 is not a divisor of 1456)
1456 / 38 = 38.315789473684 (the remainder is 12, so 38 is not a divisor of 1456)