What are the divisors of 398?

1, 2, 199, 398

There is a total of 4 positive divisors.
The sum of these divisors is 600.
The arithmetic mean is 150.

2 even divisors

2, 398

2 odd divisors

1, 199

How to compute the divisors of 398?

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 398 by each of the numbers from 1 to 398 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)

398 / 1 = 398 (the remainder is 0, so 1 is a divisor of 398)
398 / 2 = 199 (the remainder is 0, so 2 is a divisor of 398)
398 / 3 = 132.66666666667 (the remainder is 2, so 3 is not a divisor of 398)
...
398 / 397 = 1.0025188916877 (the remainder is 1, so 397 is not a divisor of 398)
398 / 398 = 1 (the remainder is 0, so 398 is a divisor of 398)

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 398 (i.e. 19.94993734326). 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 $\frac{N}{d} = D$ )

398 / 1 = 398 (the remainder is 0, so 1 and 398 are divisors of 398)
398 / 2 = 199 (the remainder is 0, so 2 and 199 are divisors of 398)
398 / 3 = 132.66666666667 (the remainder is 2, so 3 is not a divisor of 398)
...
398 / 18 = 22.111111111111 (the remainder is 2, so 18 is not a divisor of 398)
398 / 19 = 20.947368421053 (the remainder is 18, so 19 is not a divisor of 398)