What are the divisors of 3915?

1, 3, 5, 9, 15, 27, 29, 45, 87, 135, 145, 261, 435, 783, 1305, 3915

There is a total of 16 positive divisors.
The sum of these divisors is 7200.
The arithmetic mean is 450.

16 odd divisors

1, 3, 5, 9, 15, 27, 29, 45, 87, 135, 145, 261, 435, 783, 1305, 3915

How to compute the divisors of 3915?

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 3915 by each of the numbers from 1 to 3915 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)

3915 / 1 = 3915 (the remainder is 0, so 1 is a divisor of 3915)
3915 / 2 = 1957.5 (the remainder is 1, so 2 is not a divisor of 3915)
3915 / 3 = 1305 (the remainder is 0, so 3 is a divisor of 3915)
...
3915 / 3914 = 1.0002554931017 (the remainder is 1, so 3914 is not a divisor of 3915)
3915 / 3915 = 1 (the remainder is 0, so 3915 is a divisor of 3915)

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 3915 (i.e. 62.569960843843). 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 $\frac{N}{d} = D$ )

3915 / 1 = 3915 (the remainder is 0, so 1 and 3915 are divisors of 3915)
3915 / 2 = 1957.5 (the remainder is 1, so 2 is not a divisor of 3915)
3915 / 3 = 1305 (the remainder is 0, so 3 and 1305 are divisors of 3915)
...
3915 / 61 = 64.180327868852 (the remainder is 11, so 61 is not a divisor of 3915)
3915 / 62 = 63.145161290323 (the remainder is 9, so 62 is not a divisor of 3915)