What are the divisors of 1861?

1, 1861

There is a total of 2 positive divisors.
The sum of these divisors is 1862.
The arithmetic mean is 931.

2 odd divisors

1, 1861

How to compute the divisors of 1861?

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 1861 by each of the numbers from 1 to 1861 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)

1861 / 1 = 1861 (the remainder is 0, so 1 is a divisor of 1861)
1861 / 2 = 930.5 (the remainder is 1, so 2 is not a divisor of 1861)
1861 / 3 = 620.33333333333 (the remainder is 1, so 3 is not a divisor of 1861)
...
1861 / 1860 = 1.0005376344086 (the remainder is 1, so 1860 is not a divisor of 1861)
1861 / 1861 = 1 (the remainder is 0, so 1861 is a divisor of 1861)

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 1861 (i.e. 43.139309220246). 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 $\frac{N}{d} = D$ )

1861 / 1 = 1861 (the remainder is 0, so 1 and 1861 are divisors of 1861)
1861 / 2 = 930.5 (the remainder is 1, so 2 is not a divisor of 1861)
1861 / 3 = 620.33333333333 (the remainder is 1, so 3 is not a divisor of 1861)
...
1861 / 42 = 44.309523809524 (the remainder is 13, so 42 is not a divisor of 1861)
1861 / 43 = 43.279069767442 (the remainder is 12, so 43 is not a divisor of 1861)