What are the divisors of 3184?

1, 2, 4, 8, 16, 199, 398, 796, 1592, 3184

There is a total of 10 positive divisors.
The sum of these divisors is 6200.
The arithmetic mean is 620.

8 even divisors

2, 4, 8, 16, 398, 796, 1592, 3184

2 odd divisors

1, 199

How to compute the divisors of 3184?

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 3184 by each of the numbers from 1 to 3184 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)

3184 / 1 = 3184 (the remainder is 0, so 1 is a divisor of 3184)
3184 / 2 = 1592 (the remainder is 0, so 2 is a divisor of 3184)
3184 / 3 = 1061.3333333333 (the remainder is 1, so 3 is not a divisor of 3184)
...
3184 / 3183 = 1.0003141690229 (the remainder is 1, so 3183 is not a divisor of 3184)
3184 / 3184 = 1 (the remainder is 0, so 3184 is a divisor of 3184)

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 3184 (i.e. 56.426943918664). 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 $\frac{N}{d} = D$ )

3184 / 1 = 3184 (the remainder is 0, so 1 and 3184 are divisors of 3184)
3184 / 2 = 1592 (the remainder is 0, so 2 and 1592 are divisors of 3184)
3184 / 3 = 1061.3333333333 (the remainder is 1, so 3 is not a divisor of 3184)
...
3184 / 55 = 57.890909090909 (the remainder is 49, so 55 is not a divisor of 3184)
3184 / 56 = 56.857142857143 (the remainder is 48, so 56 is not a divisor of 3184)