What are the divisors of 407?

1, 11, 37, 407

There is a total of 4 positive divisors.
The sum of these divisors is 456.
The arithmetic mean is 114.

4 odd divisors

1, 11, 37, 407

How to compute the divisors of 407?

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 407 by each of the numbers from 1 to 407 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)

407 / 1 = 407 (the remainder is 0, so 1 is a divisor of 407)
407 / 2 = 203.5 (the remainder is 1, so 2 is not a divisor of 407)
407 / 3 = 135.66666666667 (the remainder is 2, so 3 is not a divisor of 407)
...
407 / 406 = 1.0024630541872 (the remainder is 1, so 406 is not a divisor of 407)
407 / 407 = 1 (the remainder is 0, so 407 is a divisor of 407)

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 407 (i.e. 20.174241001832). 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 $\frac{N}{d} = D$ )

407 / 1 = 407 (the remainder is 0, so 1 and 407 are divisors of 407)
407 / 2 = 203.5 (the remainder is 1, so 2 is not a divisor of 407)
407 / 3 = 135.66666666667 (the remainder is 2, so 3 is not a divisor of 407)
...
407 / 19 = 21.421052631579 (the remainder is 8, so 19 is not a divisor of 407)
407 / 20 = 20.35 (the remainder is 7, so 20 is not a divisor of 407)