What are the divisors of 6174?

1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174

There is a total of 24 positive divisors.
The sum of these divisors is 15600.
The arithmetic mean is 650.

12 even divisors

2, 6, 14, 18, 42, 98, 126, 294, 686, 882, 2058, 6174

12 odd divisors

1, 3, 7, 9, 21, 49, 63, 147, 343, 441, 1029, 3087

How to compute the divisors of 6174?

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 6174 by each of the numbers from 1 to 6174 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)

6174 / 1 = 6174 (the remainder is 0, so 1 is a divisor of 6174)
6174 / 2 = 3087 (the remainder is 0, so 2 is a divisor of 6174)
6174 / 3 = 2058 (the remainder is 0, so 3 is a divisor of 6174)
...
6174 / 6173 = 1.0001619957881 (the remainder is 1, so 6173 is not a divisor of 6174)
6174 / 6174 = 1 (the remainder is 0, so 6174 is a divisor of 6174)

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 6174 (i.e. 78.574805122253). 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 $\frac{N}{d} = D$ )

6174 / 1 = 6174 (the remainder is 0, so 1 and 6174 are divisors of 6174)
6174 / 2 = 3087 (the remainder is 0, so 2 and 3087 are divisors of 6174)
6174 / 3 = 2058 (the remainder is 0, so 3 and 2058 are divisors of 6174)
...
6174 / 77 = 80.181818181818 (the remainder is 14, so 77 is not a divisor of 6174)
6174 / 78 = 79.153846153846 (the remainder is 12, so 78 is not a divisor of 6174)