What are the divisors of 60?

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

There is a total of 12 positive divisors.
The sum of these divisors is 168.
The arithmetic mean is 14.

8 even divisors

2, 4, 6, 10, 12, 20, 30, 60

4 odd divisors

1, 3, 5, 15

How to compute the divisors of 60?

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 60 by each of the numbers from 1 to 60 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)

60 / 1 = 60 (the remainder is 0, so 1 is a divisor of 60)
60 / 2 = 30 (the remainder is 0, so 2 is a divisor of 60)
60 / 3 = 20 (the remainder is 0, so 3 is a divisor of 60)
...
60 / 59 = 1.0169491525424 (the remainder is 1, so 59 is not a divisor of 60)
60 / 60 = 1 (the remainder is 0, so 60 is a divisor of 60)

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 60 (i.e. 7.7459666924148). 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 $\frac{N}{d} = D$ )

60 / 1 = 60 (the remainder is 0, so 1 and 60 are divisors of 60)
60 / 2 = 30 (the remainder is 0, so 2 and 30 are divisors of 60)
60 / 3 = 20 (the remainder is 0, so 3 and 20 are divisors of 60)
...
60 / 6 = 10 (the remainder is 0, so 6 and 10 are divisors of 60)
60 / 7 = 8.5714285714286 (the remainder is 4, so 7 is not a divisor of 60)