What are the divisors of 692?

1, 2, 4, 173, 346, 692

There is a total of 6 positive divisors.
The sum of these divisors is 1218.
The arithmetic mean is 203.

4 even divisors

2, 4, 346, 692

2 odd divisors

1, 173

How to compute the divisors of 692?

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 692 by each of the numbers from 1 to 692 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)

692 / 1 = 692 (the remainder is 0, so 1 is a divisor of 692)
692 / 2 = 346 (the remainder is 0, so 2 is a divisor of 692)
692 / 3 = 230.66666666667 (the remainder is 2, so 3 is not a divisor of 692)
...
692 / 691 = 1.0014471780029 (the remainder is 1, so 691 is not a divisor of 692)
692 / 692 = 1 (the remainder is 0, so 692 is a divisor of 692)

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 692 (i.e. 26.305892875932). 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 $\frac{N}{d} = D$ )

692 / 1 = 692 (the remainder is 0, so 1 and 692 are divisors of 692)
692 / 2 = 346 (the remainder is 0, so 2 and 346 are divisors of 692)
692 / 3 = 230.66666666667 (the remainder is 2, so 3 is not a divisor of 692)
...
692 / 25 = 27.68 (the remainder is 17, so 25 is not a divisor of 692)
692 / 26 = 26.615384615385 (the remainder is 16, so 26 is not a divisor of 692)