What are the divisors of 417?

1, 3, 139, 417

There is a total of 4 positive divisors.
The sum of these divisors is 560.
The arithmetic mean is 140.

4 odd divisors

1, 3, 139, 417

How to compute the divisors of 417?

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 417 by each of the numbers from 1 to 417 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)

417 / 1 = 417 (the remainder is 0, so 1 is a divisor of 417)
417 / 2 = 208.5 (the remainder is 1, so 2 is not a divisor of 417)
417 / 3 = 139 (the remainder is 0, so 3 is a divisor of 417)
...
417 / 416 = 1.0024038461538 (the remainder is 1, so 416 is not a divisor of 417)
417 / 417 = 1 (the remainder is 0, so 417 is a divisor of 417)

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 417 (i.e. 20.420577856662). 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 $\frac{N}{d} = D$ )

417 / 1 = 417 (the remainder is 0, so 1 and 417 are divisors of 417)
417 / 2 = 208.5 (the remainder is 1, so 2 is not a divisor of 417)
417 / 3 = 139 (the remainder is 0, so 3 and 139 are divisors of 417)
...
417 / 19 = 21.947368421053 (the remainder is 18, so 19 is not a divisor of 417)
417 / 20 = 20.85 (the remainder is 17, so 20 is not a divisor of 417)