What are the divisors of 8174?

1, 2, 61, 67, 122, 134, 4087, 8174

There is a total of 8 positive divisors.
The sum of these divisors is 12648.
The arithmetic mean is 1581.

4 even divisors

2, 122, 134, 8174

4 odd divisors

1, 61, 67, 4087

How to compute the divisors of 8174?

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 8174 by each of the numbers from 1 to 8174 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)

8174 / 1 = 8174 (the remainder is 0, so 1 is a divisor of 8174)
8174 / 2 = 4087 (the remainder is 0, so 2 is a divisor of 8174)
8174 / 3 = 2724.6666666667 (the remainder is 2, so 3 is not a divisor of 8174)
...
8174 / 8173 = 1.0001223540927 (the remainder is 1, so 8173 is not a divisor of 8174)
8174 / 8174 = 1 (the remainder is 0, so 8174 is a divisor of 8174)

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 8174 (i.e. 90.410176418366). 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 $\frac{N}{d} = D$ )

8174 / 1 = 8174 (the remainder is 0, so 1 and 8174 are divisors of 8174)
8174 / 2 = 4087 (the remainder is 0, so 2 and 4087 are divisors of 8174)
8174 / 3 = 2724.6666666667 (the remainder is 2, so 3 is not a divisor of 8174)
...
8174 / 89 = 91.842696629213 (the remainder is 75, so 89 is not a divisor of 8174)
8174 / 90 = 90.822222222222 (the remainder is 74, so 90 is not a divisor of 8174)