What are the divisors of 1991?

1, 11, 181, 1991

There is a total of 4 positive divisors.
The sum of these divisors is 2184.
The arithmetic mean is 546.

4 odd divisors

1, 11, 181, 1991

How to compute the divisors of 1991?

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 1991 by each of the numbers from 1 to 1991 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)

1991 / 1 = 1991 (the remainder is 0, so 1 is a divisor of 1991)
1991 / 2 = 995.5 (the remainder is 1, so 2 is not a divisor of 1991)
1991 / 3 = 663.66666666667 (the remainder is 2, so 3 is not a divisor of 1991)
...
1991 / 1990 = 1.0005025125628 (the remainder is 1, so 1990 is not a divisor of 1991)
1991 / 1991 = 1 (the remainder is 0, so 1991 is a divisor of 1991)

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 1991 (i.e. 44.620623034646). 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 $\frac{N}{d} = D$ )

1991 / 1 = 1991 (the remainder is 0, so 1 and 1991 are divisors of 1991)
1991 / 2 = 995.5 (the remainder is 1, so 2 is not a divisor of 1991)
1991 / 3 = 663.66666666667 (the remainder is 2, so 3 is not a divisor of 1991)
...
1991 / 43 = 46.302325581395 (the remainder is 13, so 43 is not a divisor of 1991)
1991 / 44 = 45.25 (the remainder is 11, so 44 is not a divisor of 1991)