What are the divisors of 3720?

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 31, 40, 60, 62, 93, 120, 124, 155, 186, 248, 310, 372, 465, 620, 744, 930, 1240, 1860, 3720

There is a total of 32 positive divisors.
The sum of these divisors is 11520.
The arithmetic mean is 360.

24 even divisors

2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 62, 120, 124, 186, 248, 310, 372, 620, 744, 930, 1240, 1860, 3720

8 odd divisors

1, 3, 5, 15, 31, 93, 155, 465

How to compute the divisors of 3720?

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 3720 by each of the numbers from 1 to 3720 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)

3720 / 1 = 3720 (the remainder is 0, so 1 is a divisor of 3720)
3720 / 2 = 1860 (the remainder is 0, so 2 is a divisor of 3720)
3720 / 3 = 1240 (the remainder is 0, so 3 is a divisor of 3720)
...
3720 / 3719 = 1.0002688894864 (the remainder is 1, so 3719 is not a divisor of 3720)
3720 / 3720 = 1 (the remainder is 0, so 3720 is a divisor of 3720)

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 3720 (i.e. 60.991802727908). 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 $\frac{N}{d} = D$ )

3720 / 1 = 3720 (the remainder is 0, so 1 and 3720 are divisors of 3720)
3720 / 2 = 1860 (the remainder is 0, so 2 and 1860 are divisors of 3720)
3720 / 3 = 1240 (the remainder is 0, so 3 and 1240 are divisors of 3720)
...
3720 / 59 = 63.050847457627 (the remainder is 3, so 59 is not a divisor of 3720)
3720 / 60 = 62 (the remainder is 0, so 60 and 62 are divisors of 3720)