What are the divisors of 411?

1, 3, 137, 411

There is a total of 4 positive divisors.
The sum of these divisors is 552.
The arithmetic mean is 138.

4 odd divisors

1, 3, 137, 411

How to compute the divisors of 411?

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 411 by each of the numbers from 1 to 411 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)

411 / 1 = 411 (the remainder is 0, so 1 is a divisor of 411)
411 / 2 = 205.5 (the remainder is 1, so 2 is not a divisor of 411)
411 / 3 = 137 (the remainder is 0, so 3 is a divisor of 411)
...
411 / 410 = 1.0024390243902 (the remainder is 1, so 410 is not a divisor of 411)
411 / 411 = 1 (the remainder is 0, so 411 is a divisor of 411)

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 411 (i.e. 20.273134932713). 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 $\frac{N}{d} = D$ )

411 / 1 = 411 (the remainder is 0, so 1 and 411 are divisors of 411)
411 / 2 = 205.5 (the remainder is 1, so 2 is not a divisor of 411)
411 / 3 = 137 (the remainder is 0, so 3 and 137 are divisors of 411)
...
411 / 19 = 21.631578947368 (the remainder is 12, so 19 is not a divisor of 411)
411 / 20 = 20.55 (the remainder is 11, so 20 is not a divisor of 411)