What are the divisors of 1945?

1, 5, 389, 1945

There is a total of 4 positive divisors.
The sum of these divisors is 2340.
The arithmetic mean is 585.

4 odd divisors

1, 5, 389, 1945

How to compute the divisors of 1945?

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 1945 by each of the numbers from 1 to 1945 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)

1945 / 1 = 1945 (the remainder is 0, so 1 is a divisor of 1945)
1945 / 2 = 972.5 (the remainder is 1, so 2 is not a divisor of 1945)
1945 / 3 = 648.33333333333 (the remainder is 1, so 3 is not a divisor of 1945)
...
1945 / 1944 = 1.0005144032922 (the remainder is 1, so 1944 is not a divisor of 1945)
1945 / 1945 = 1 (the remainder is 0, so 1945 is a divisor of 1945)

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 1945 (i.e. 44.1021541424). 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 $\frac{N}{d} = D$ )

1945 / 1 = 1945 (the remainder is 0, so 1 and 1945 are divisors of 1945)
1945 / 2 = 972.5 (the remainder is 1, so 2 is not a divisor of 1945)
1945 / 3 = 648.33333333333 (the remainder is 1, so 3 is not a divisor of 1945)
...
1945 / 43 = 45.232558139535 (the remainder is 10, so 43 is not a divisor of 1945)
1945 / 44 = 44.204545454545 (the remainder is 9, so 44 is not a divisor of 1945)