What are the divisors of 1696?

1, 2, 4, 8, 16, 32, 53, 106, 212, 424, 848, 1696

There is a total of 12 positive divisors.
The sum of these divisors is 3402.
The arithmetic mean is 283.5.

10 even divisors

2, 4, 8, 16, 32, 106, 212, 424, 848, 1696

2 odd divisors

1, 53

How to compute the divisors of 1696?

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 1696 by each of the numbers from 1 to 1696 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)

1696 / 1 = 1696 (the remainder is 0, so 1 is a divisor of 1696)
1696 / 2 = 848 (the remainder is 0, so 2 is a divisor of 1696)
1696 / 3 = 565.33333333333 (the remainder is 1, so 3 is not a divisor of 1696)
...
1696 / 1695 = 1.0005899705015 (the remainder is 1, so 1695 is not a divisor of 1696)
1696 / 1696 = 1 (the remainder is 0, so 1696 is a divisor of 1696)

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 1696 (i.e. 41.182520563948). 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 $\frac{N}{d} = D$ )

1696 / 1 = 1696 (the remainder is 0, so 1 and 1696 are divisors of 1696)
1696 / 2 = 848 (the remainder is 0, so 2 and 848 are divisors of 1696)
1696 / 3 = 565.33333333333 (the remainder is 1, so 3 is not a divisor of 1696)
...
1696 / 40 = 42.4 (the remainder is 16, so 40 is not a divisor of 1696)
1696 / 41 = 41.365853658537 (the remainder is 15, so 41 is not a divisor of 1696)