What are the divisors of 4991?

1, 7, 23, 31, 161, 217, 713, 4991

There is a total of 8 positive divisors.
The sum of these divisors is 6144.
The arithmetic mean is 768.

8 odd divisors

1, 7, 23, 31, 161, 217, 713, 4991

How to compute the divisors of 4991?

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 4991 by each of the numbers from 1 to 4991 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)

4991 / 1 = 4991 (the remainder is 0, so 1 is a divisor of 4991)
4991 / 2 = 2495.5 (the remainder is 1, so 2 is not a divisor of 4991)
4991 / 3 = 1663.6666666667 (the remainder is 2, so 3 is not a divisor of 4991)
...
4991 / 4990 = 1.0002004008016 (the remainder is 1, so 4990 is not a divisor of 4991)
4991 / 4991 = 1 (the remainder is 0, so 4991 is a divisor of 4991)

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 4991 (i.e. 70.64700984472). 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 $\frac{N}{d} = D$ )

4991 / 1 = 4991 (the remainder is 0, so 1 and 4991 are divisors of 4991)
4991 / 2 = 2495.5 (the remainder is 1, so 2 is not a divisor of 4991)
4991 / 3 = 1663.6666666667 (the remainder is 2, so 3 is not a divisor of 4991)
...
4991 / 69 = 72.333333333333 (the remainder is 23, so 69 is not a divisor of 4991)
4991 / 70 = 71.3 (the remainder is 21, so 70 is not a divisor of 4991)