What are the divisors of 1919?

1, 19, 101, 1919

There is a total of 4 positive divisors.
The sum of these divisors is 2040.
The arithmetic mean is 510.

4 odd divisors

1, 19, 101, 1919

How to compute the divisors of 1919?

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 1919 by each of the numbers from 1 to 1919 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)

1919 / 1 = 1919 (the remainder is 0, so 1 is a divisor of 1919)
1919 / 2 = 959.5 (the remainder is 1, so 2 is not a divisor of 1919)
1919 / 3 = 639.66666666667 (the remainder is 2, so 3 is not a divisor of 1919)
...
1919 / 1918 = 1.0005213764338 (the remainder is 1, so 1918 is not a divisor of 1919)
1919 / 1919 = 1 (the remainder is 0, so 1919 is a divisor of 1919)

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 1919 (i.e. 43.806392227619). 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 $\frac{N}{d} = D$ )

1919 / 1 = 1919 (the remainder is 0, so 1 and 1919 are divisors of 1919)
1919 / 2 = 959.5 (the remainder is 1, so 2 is not a divisor of 1919)
1919 / 3 = 639.66666666667 (the remainder is 2, so 3 is not a divisor of 1919)
...
1919 / 42 = 45.690476190476 (the remainder is 29, so 42 is not a divisor of 1919)
1919 / 43 = 44.627906976744 (the remainder is 27, so 43 is not a divisor of 1919)