What are the divisors of 1682?

1, 2, 29, 58, 841, 1682

There is a total of 6 positive divisors.
The sum of these divisors is 2613.
The arithmetic mean is 435.5.

3 even divisors

2, 58, 1682

3 odd divisors

1, 29, 841

How to compute the divisors of 1682?

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 1682 by each of the numbers from 1 to 1682 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)

1682 / 1 = 1682 (the remainder is 0, so 1 is a divisor of 1682)
1682 / 2 = 841 (the remainder is 0, so 2 is a divisor of 1682)
1682 / 3 = 560.66666666667 (the remainder is 2, so 3 is not a divisor of 1682)
...
1682 / 1681 = 1.0005948839976 (the remainder is 1, so 1681 is not a divisor of 1682)
1682 / 1682 = 1 (the remainder is 0, so 1682 is a divisor of 1682)

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 1682 (i.e. 41.01219330882). 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 $\frac{N}{d} = D$ )

1682 / 1 = 1682 (the remainder is 0, so 1 and 1682 are divisors of 1682)
1682 / 2 = 841 (the remainder is 0, so 2 and 841 are divisors of 1682)
1682 / 3 = 560.66666666667 (the remainder is 2, so 3 is not a divisor of 1682)
...
1682 / 40 = 42.05 (the remainder is 2, so 40 is not a divisor of 1682)
1682 / 41 = 41.024390243902 (the remainder is 1, so 41 is not a divisor of 1682)