What are the divisors of 919?

1, 919

There is a total of 2 positive divisors.
The sum of these divisors is 920.
The arithmetic mean is 460.

2 odd divisors

1, 919

How to compute the divisors of 919?

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 919 by each of the numbers from 1 to 919 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)

919 / 1 = 919 (the remainder is 0, so 1 is a divisor of 919)
919 / 2 = 459.5 (the remainder is 1, so 2 is not a divisor of 919)
919 / 3 = 306.33333333333 (the remainder is 1, so 3 is not a divisor of 919)
...
919 / 918 = 1.0010893246187 (the remainder is 1, so 918 is not a divisor of 919)
919 / 919 = 1 (the remainder is 0, so 919 is a divisor of 919)

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 919 (i.e. 30.315012782448). 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 $\frac{N}{d} = D$ )

919 / 1 = 919 (the remainder is 0, so 1 and 919 are divisors of 919)
919 / 2 = 459.5 (the remainder is 1, so 2 is not a divisor of 919)
919 / 3 = 306.33333333333 (the remainder is 1, so 3 is not a divisor of 919)
...
919 / 29 = 31.689655172414 (the remainder is 20, so 29 is not a divisor of 919)
919 / 30 = 30.633333333333 (the remainder is 19, so 30 is not a divisor of 919)