What are the divisors of 416?

1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416

There is a total of 12 positive divisors.
The sum of these divisors is 882.
The arithmetic mean is 73.5.

10 even divisors

2, 4, 8, 16, 26, 32, 52, 104, 208, 416

2 odd divisors

1, 13

How to compute the divisors of 416?

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 416 by each of the numbers from 1 to 416 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)

416 / 1 = 416 (the remainder is 0, so 1 is a divisor of 416)
416 / 2 = 208 (the remainder is 0, so 2 is a divisor of 416)
416 / 3 = 138.66666666667 (the remainder is 2, so 3 is not a divisor of 416)
...
416 / 415 = 1.0024096385542 (the remainder is 1, so 415 is not a divisor of 416)
416 / 416 = 1 (the remainder is 0, so 416 is a divisor of 416)

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 416 (i.e. 20.396078054371). 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 $\frac{N}{d} = D$ )

416 / 1 = 416 (the remainder is 0, so 1 and 416 are divisors of 416)
416 / 2 = 208 (the remainder is 0, so 2 and 208 are divisors of 416)
416 / 3 = 138.66666666667 (the remainder is 2, so 3 is not a divisor of 416)
...
416 / 19 = 21.894736842105 (the remainder is 17, so 19 is not a divisor of 416)
416 / 20 = 20.8 (the remainder is 16, so 20 is not a divisor of 416)