What are the divisors of 329?

1, 7, 47, 329

There is a total of 4 positive divisors.
The sum of these divisors is 384.
The arithmetic mean is 96.

4 odd divisors

1, 7, 47, 329

How to compute the divisors of 329?

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 329 by each of the numbers from 1 to 329 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)

329 / 1 = 329 (the remainder is 0, so 1 is a divisor of 329)
329 / 2 = 164.5 (the remainder is 1, so 2 is not a divisor of 329)
329 / 3 = 109.66666666667 (the remainder is 2, so 3 is not a divisor of 329)
...
329 / 328 = 1.0030487804878 (the remainder is 1, so 328 is not a divisor of 329)
329 / 329 = 1 (the remainder is 0, so 329 is a divisor of 329)

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 329 (i.e. 18.138357147217). 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 $\frac{N}{d} = D$ )

329 / 1 = 329 (the remainder is 0, so 1 and 329 are divisors of 329)
329 / 2 = 164.5 (the remainder is 1, so 2 is not a divisor of 329)
329 / 3 = 109.66666666667 (the remainder is 2, so 3 is not a divisor of 329)
...
329 / 17 = 19.352941176471 (the remainder is 6, so 17 is not a divisor of 329)
329 / 18 = 18.277777777778 (the remainder is 5, so 18 is not a divisor of 329)