What are the divisors of 1408?

1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 128, 176, 352, 704, 1408

There is a total of 16 positive divisors.
The sum of these divisors is 3060.
The arithmetic mean is 191.25.

14 even divisors

2, 4, 8, 16, 22, 32, 44, 64, 88, 128, 176, 352, 704, 1408

2 odd divisors

1, 11

How to compute the divisors of 1408?

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 1408 by each of the numbers from 1 to 1408 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)

1408 / 1 = 1408 (the remainder is 0, so 1 is a divisor of 1408)
1408 / 2 = 704 (the remainder is 0, so 2 is a divisor of 1408)
1408 / 3 = 469.33333333333 (the remainder is 1, so 3 is not a divisor of 1408)
...
1408 / 1407 = 1.000710732054 (the remainder is 1, so 1407 is not a divisor of 1408)
1408 / 1408 = 1 (the remainder is 0, so 1408 is a divisor of 1408)

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 1408 (i.e. 37.523326078587). 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 $\frac{N}{d} = D$ )

1408 / 1 = 1408 (the remainder is 0, so 1 and 1408 are divisors of 1408)
1408 / 2 = 704 (the remainder is 0, so 2 and 704 are divisors of 1408)
1408 / 3 = 469.33333333333 (the remainder is 1, so 3 is not a divisor of 1408)
...
1408 / 36 = 39.111111111111 (the remainder is 4, so 36 is not a divisor of 1408)
1408 / 37 = 38.054054054054 (the remainder is 2, so 37 is not a divisor of 1408)