What are the divisors of 345?

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

There is a total of 8 positive divisors.
The sum of these divisors is 576.
The arithmetic mean is 72.

8 odd divisors

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

How to compute the divisors of 345?

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 345 by each of the numbers from 1 to 345 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)

345 / 1 = 345 (the remainder is 0, so 1 is a divisor of 345)
345 / 2 = 172.5 (the remainder is 1, so 2 is not a divisor of 345)
345 / 3 = 115 (the remainder is 0, so 3 is a divisor of 345)
...
345 / 344 = 1.0029069767442 (the remainder is 1, so 344 is not a divisor of 345)
345 / 345 = 1 (the remainder is 0, so 345 is a divisor of 345)

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 345 (i.e. 18.574175621007). 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 $\frac{N}{d} = D$ )

345 / 1 = 345 (the remainder is 0, so 1 and 345 are divisors of 345)
345 / 2 = 172.5 (the remainder is 1, so 2 is not a divisor of 345)
345 / 3 = 115 (the remainder is 0, so 3 and 115 are divisors of 345)
...
345 / 17 = 20.294117647059 (the remainder is 5, so 17 is not a divisor of 345)
345 / 18 = 19.166666666667 (the remainder is 3, so 18 is not a divisor of 345)