What are the divisors of 821?

1, 821

There is a total of 2 positive divisors.
The sum of these divisors is 822.
The arithmetic mean is 411.

2 odd divisors

1, 821

How to compute the divisors of 821?

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 821 by each of the numbers from 1 to 821 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)

821 / 1 = 821 (the remainder is 0, so 1 is a divisor of 821)
821 / 2 = 410.5 (the remainder is 1, so 2 is not a divisor of 821)
821 / 3 = 273.66666666667 (the remainder is 2, so 3 is not a divisor of 821)
...
821 / 820 = 1.0012195121951 (the remainder is 1, so 820 is not a divisor of 821)
821 / 821 = 1 (the remainder is 0, so 821 is a divisor of 821)

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 821 (i.e. 28.653097563789). 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 $\frac{N}{d} = D$ )

821 / 1 = 821 (the remainder is 0, so 1 and 821 are divisors of 821)
821 / 2 = 410.5 (the remainder is 1, so 2 is not a divisor of 821)
821 / 3 = 273.66666666667 (the remainder is 2, so 3 is not a divisor of 821)
...
821 / 27 = 30.407407407407 (the remainder is 11, so 27 is not a divisor of 821)
821 / 28 = 29.321428571429 (the remainder is 9, so 28 is not a divisor of 821)