What are the divisors of 1688?

1, 2, 4, 8, 211, 422, 844, 1688

There is a total of 8 positive divisors.
The sum of these divisors is 3180.
The arithmetic mean is 397.5.

6 even divisors

2, 4, 8, 422, 844, 1688

2 odd divisors

1, 211

How to compute the divisors of 1688?

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 1688 by each of the numbers from 1 to 1688 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)

1688 / 1 = 1688 (the remainder is 0, so 1 is a divisor of 1688)
1688 / 2 = 844 (the remainder is 0, so 2 is a divisor of 1688)
1688 / 3 = 562.66666666667 (the remainder is 2, so 3 is not a divisor of 1688)
...
1688 / 1687 = 1.0005927682276 (the remainder is 1, so 1687 is not a divisor of 1688)
1688 / 1688 = 1 (the remainder is 0, so 1688 is a divisor of 1688)

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 1688 (i.e. 41.085277168348). 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 $\frac{N}{d} = D$ )

1688 / 1 = 1688 (the remainder is 0, so 1 and 1688 are divisors of 1688)
1688 / 2 = 844 (the remainder is 0, so 2 and 844 are divisors of 1688)
1688 / 3 = 562.66666666667 (the remainder is 2, so 3 is not a divisor of 1688)
...
1688 / 40 = 42.2 (the remainder is 8, so 40 is not a divisor of 1688)
1688 / 41 = 41.170731707317 (the remainder is 7, so 41 is not a divisor of 1688)