What are the divisors of 827?

1, 827

There is a total of 2 positive divisors.
The sum of these divisors is 828.
The arithmetic mean is 414.

2 odd divisors

1, 827

How to compute the divisors of 827?

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 827 by each of the numbers from 1 to 827 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)

827 / 1 = 827 (the remainder is 0, so 1 is a divisor of 827)
827 / 2 = 413.5 (the remainder is 1, so 2 is not a divisor of 827)
827 / 3 = 275.66666666667 (the remainder is 2, so 3 is not a divisor of 827)
...
827 / 826 = 1.001210653753 (the remainder is 1, so 826 is not a divisor of 827)
827 / 827 = 1 (the remainder is 0, so 827 is a divisor of 827)

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 827 (i.e. 28.757607689097). 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 $\frac{N}{d} = D$ )

827 / 1 = 827 (the remainder is 0, so 1 and 827 are divisors of 827)
827 / 2 = 413.5 (the remainder is 1, so 2 is not a divisor of 827)
827 / 3 = 275.66666666667 (the remainder is 2, so 3 is not a divisor of 827)
...
827 / 27 = 30.62962962963 (the remainder is 17, so 27 is not a divisor of 827)
827 / 28 = 29.535714285714 (the remainder is 15, so 28 is not a divisor of 827)