What are the divisors of 419?

1, 419

There is a total of 2 positive divisors.
The sum of these divisors is 420.
The arithmetic mean is 210.

2 odd divisors

1, 419

How to compute the divisors of 419?

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 419 by each of the numbers from 1 to 419 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)

419 / 1 = 419 (the remainder is 0, so 1 is a divisor of 419)
419 / 2 = 209.5 (the remainder is 1, so 2 is not a divisor of 419)
419 / 3 = 139.66666666667 (the remainder is 2, so 3 is not a divisor of 419)
...
419 / 418 = 1.0023923444976 (the remainder is 1, so 418 is not a divisor of 419)
419 / 419 = 1 (the remainder is 0, so 419 is a divisor of 419)

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 419 (i.e. 20.469489490459). 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 $\frac{N}{d} = D$ )

419 / 1 = 419 (the remainder is 0, so 1 and 419 are divisors of 419)
419 / 2 = 209.5 (the remainder is 1, so 2 is not a divisor of 419)
419 / 3 = 139.66666666667 (the remainder is 2, so 3 is not a divisor of 419)
...
419 / 19 = 22.052631578947 (the remainder is 1, so 19 is not a divisor of 419)
419 / 20 = 20.95 (the remainder is 19, so 20 is not a divisor of 419)