What are the divisors of 841?

1, 29, 841

There is a total of 3 positive divisors.
The sum of these divisors is 871.
The arithmetic mean is 290.33333333333.

3 odd divisors

1, 29, 841

How to compute the divisors of 841?

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 841 by each of the numbers from 1 to 841 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)

841 / 1 = 841 (the remainder is 0, so 1 is a divisor of 841)
841 / 2 = 420.5 (the remainder is 1, so 2 is not a divisor of 841)
841 / 3 = 280.33333333333 (the remainder is 1, so 3 is not a divisor of 841)
...
841 / 840 = 1.0011904761905 (the remainder is 1, so 840 is not a divisor of 841)
841 / 841 = 1 (the remainder is 0, so 841 is a divisor of 841)

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 841 (i.e. 29). 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 $\frac{N}{d} = D$ )

841 / 1 = 841 (the remainder is 0, so 1 and 841 are divisors of 841)
841 / 2 = 420.5 (the remainder is 1, so 2 is not a divisor of 841)
841 / 3 = 280.33333333333 (the remainder is 1, so 3 is not a divisor of 841)
...
841 / 28 = 30.035714285714 (the remainder is 1, so 28 is not a divisor of 841)
841 / 29 = 29 (the remainder is 0, so 29 and 29 are divisors of 841)