What are the divisors of 690?

1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690

There is a total of 16 positive divisors.
The sum of these divisors is 1728.
The arithmetic mean is 108.

8 even divisors

2, 6, 10, 30, 46, 138, 230, 690

8 odd divisors

1, 3, 5, 15, 23, 69, 115, 345

How to compute the divisors of 690?

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 690 by each of the numbers from 1 to 690 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)

690 / 1 = 690 (the remainder is 0, so 1 is a divisor of 690)
690 / 2 = 345 (the remainder is 0, so 2 is a divisor of 690)
690 / 3 = 230 (the remainder is 0, so 3 is a divisor of 690)
...
690 / 689 = 1.0014513788099 (the remainder is 1, so 689 is not a divisor of 690)
690 / 690 = 1 (the remainder is 0, so 690 is a divisor of 690)

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 690 (i.e. 26.267851073127). 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 $\frac{N}{d} = D$ )

690 / 1 = 690 (the remainder is 0, so 1 and 690 are divisors of 690)
690 / 2 = 345 (the remainder is 0, so 2 and 345 are divisors of 690)
690 / 3 = 230 (the remainder is 0, so 3 and 230 are divisors of 690)
...
690 / 25 = 27.6 (the remainder is 15, so 25 is not a divisor of 690)
690 / 26 = 26.538461538462 (the remainder is 14, so 26 is not a divisor of 690)