What are the divisors of 3167?

1, 3167

There is a total of 2 positive divisors.
The sum of these divisors is 3168.
The arithmetic mean is 1584.

2 odd divisors

1, 3167

How to compute the divisors of 3167?

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 3167 by each of the numbers from 1 to 3167 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)

3167 / 1 = 3167 (the remainder is 0, so 1 is a divisor of 3167)
3167 / 2 = 1583.5 (the remainder is 1, so 2 is not a divisor of 3167)
3167 / 3 = 1055.6666666667 (the remainder is 2, so 3 is not a divisor of 3167)
...
3167 / 3166 = 1.0003158559697 (the remainder is 1, so 3166 is not a divisor of 3167)
3167 / 3167 = 1 (the remainder is 0, so 3167 is a divisor of 3167)

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 3167 (i.e. 56.276105053566). 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 $\frac{N}{d} = D$ )

3167 / 1 = 3167 (the remainder is 0, so 1 and 3167 are divisors of 3167)
3167 / 2 = 1583.5 (the remainder is 1, so 2 is not a divisor of 3167)
3167 / 3 = 1055.6666666667 (the remainder is 2, so 3 is not a divisor of 3167)
...
3167 / 55 = 57.581818181818 (the remainder is 32, so 55 is not a divisor of 3167)
3167 / 56 = 56.553571428571 (the remainder is 31, so 56 is not a divisor of 3167)