What are the divisors of 3157?

1, 7, 11, 41, 77, 287, 451, 3157

There is a total of 8 positive divisors.
The sum of these divisors is 4032.
The arithmetic mean is 504.

8 odd divisors

1, 7, 11, 41, 77, 287, 451, 3157

How to compute the divisors of 3157?

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 3157 by each of the numbers from 1 to 3157 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)

3157 / 1 = 3157 (the remainder is 0, so 1 is a divisor of 3157)
3157 / 2 = 1578.5 (the remainder is 1, so 2 is not a divisor of 3157)
3157 / 3 = 1052.3333333333 (the remainder is 1, so 3 is not a divisor of 3157)
...
3157 / 3156 = 1.0003168567807 (the remainder is 1, so 3156 is not a divisor of 3157)
3157 / 3157 = 1 (the remainder is 0, so 3157 is a divisor of 3157)

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 3157 (i.e. 56.187187151521). 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 $\frac{N}{d} = D$ )

3157 / 1 = 3157 (the remainder is 0, so 1 and 3157 are divisors of 3157)
3157 / 2 = 1578.5 (the remainder is 1, so 2 is not a divisor of 3157)
3157 / 3 = 1052.3333333333 (the remainder is 1, so 3 is not a divisor of 3157)
...
3157 / 55 = 57.4 (the remainder is 22, so 55 is not a divisor of 3157)
3157 / 56 = 56.375 (the remainder is 21, so 56 is not a divisor of 3157)