What are the divisors of 1681?

1, 41, 1681

There is a total of 3 positive divisors.
The sum of these divisors is 1723.
The arithmetic mean is 574.33333333333.

3 odd divisors

1, 41, 1681

How to compute the divisors of 1681?

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 1681 by each of the numbers from 1 to 1681 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)

1681 / 1 = 1681 (the remainder is 0, so 1 is a divisor of 1681)
1681 / 2 = 840.5 (the remainder is 1, so 2 is not a divisor of 1681)
1681 / 3 = 560.33333333333 (the remainder is 1, so 3 is not a divisor of 1681)
...
1681 / 1680 = 1.0005952380952 (the remainder is 1, so 1680 is not a divisor of 1681)
1681 / 1681 = 1 (the remainder is 0, so 1681 is a divisor of 1681)

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 1681 (i.e. 41). 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 $\frac{N}{d} = D$ )

1681 / 1 = 1681 (the remainder is 0, so 1 and 1681 are divisors of 1681)
1681 / 2 = 840.5 (the remainder is 1, so 2 is not a divisor of 1681)
1681 / 3 = 560.33333333333 (the remainder is 1, so 3 is not a divisor of 1681)
...
1681 / 40 = 42.025 (the remainder is 1, so 40 is not a divisor of 1681)
1681 / 41 = 41 (the remainder is 0, so 41 and 41 are divisors of 1681)