What are the divisors of 1933?

1, 1933

There is a total of 2 positive divisors.
The sum of these divisors is 1934.
The arithmetic mean is 967.

2 odd divisors

1, 1933

How to compute the divisors of 1933?

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 1933 by each of the numbers from 1 to 1933 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)

1933 / 1 = 1933 (the remainder is 0, so 1 is a divisor of 1933)
1933 / 2 = 966.5 (the remainder is 1, so 2 is not a divisor of 1933)
1933 / 3 = 644.33333333333 (the remainder is 1, so 3 is not a divisor of 1933)
...
1933 / 1932 = 1.0005175983437 (the remainder is 1, so 1932 is not a divisor of 1933)
1933 / 1933 = 1 (the remainder is 0, so 1933 is a divisor of 1933)

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 1933 (i.e. 43.965895873961). 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 $\frac{N}{d} = D$ )

1933 / 1 = 1933 (the remainder is 0, so 1 and 1933 are divisors of 1933)
1933 / 2 = 966.5 (the remainder is 1, so 2 is not a divisor of 1933)
1933 / 3 = 644.33333333333 (the remainder is 1, so 3 is not a divisor of 1933)
...
1933 / 42 = 46.02380952381 (the remainder is 1, so 42 is not a divisor of 1933)
1933 / 43 = 44.953488372093 (the remainder is 41, so 43 is not a divisor of 1933)