What are the divisors of 693?

1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693

There is a total of 12 positive divisors.
The sum of these divisors is 1248.
The arithmetic mean is 104.

12 odd divisors

1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693

How to compute the divisors of 693?

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 693 by each of the numbers from 1 to 693 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)

693 / 1 = 693 (the remainder is 0, so 1 is a divisor of 693)
693 / 2 = 346.5 (the remainder is 1, so 2 is not a divisor of 693)
693 / 3 = 231 (the remainder is 0, so 3 is a divisor of 693)
...
693 / 692 = 1.0014450867052 (the remainder is 1, so 692 is not a divisor of 693)
693 / 693 = 1 (the remainder is 0, so 693 is a divisor of 693)

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 693 (i.e. 26.324893162176). 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 $\frac{N}{d} = D$ )

693 / 1 = 693 (the remainder is 0, so 1 and 693 are divisors of 693)
693 / 2 = 346.5 (the remainder is 1, so 2 is not a divisor of 693)
693 / 3 = 231 (the remainder is 0, so 3 and 231 are divisors of 693)
...
693 / 25 = 27.72 (the remainder is 18, so 25 is not a divisor of 693)
693 / 26 = 26.653846153846 (the remainder is 17, so 26 is not a divisor of 693)