What are the divisors of 870?

1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870

There is a total of 16 positive divisors.
The sum of these divisors is 2160.
The arithmetic mean is 135.

8 even divisors

2, 6, 10, 30, 58, 174, 290, 870

8 odd divisors

1, 3, 5, 15, 29, 87, 145, 435

How to compute the divisors of 870?

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 870 by each of the numbers from 1 to 870 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)

870 / 1 = 870 (the remainder is 0, so 1 is a divisor of 870)
870 / 2 = 435 (the remainder is 0, so 2 is a divisor of 870)
870 / 3 = 290 (the remainder is 0, so 3 is a divisor of 870)
...
870 / 869 = 1.0011507479862 (the remainder is 1, so 869 is not a divisor of 870)
870 / 870 = 1 (the remainder is 0, so 870 is a divisor of 870)

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 870 (i.e. 29.495762407505). 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 $\frac{N}{d} = D$ )

870 / 1 = 870 (the remainder is 0, so 1 and 870 are divisors of 870)
870 / 2 = 435 (the remainder is 0, so 2 and 435 are divisors of 870)
870 / 3 = 290 (the remainder is 0, so 3 and 290 are divisors of 870)
...
870 / 28 = 31.071428571429 (the remainder is 2, so 28 is not a divisor of 870)
870 / 29 = 30 (the remainder is 0, so 29 and 30 are divisors of 870)