What are the divisors of 351?

1, 3, 9, 13, 27, 39, 117, 351

There is a total of 8 positive divisors.
The sum of these divisors is 560.
The arithmetic mean is 70.

8 odd divisors

1, 3, 9, 13, 27, 39, 117, 351

How to compute the divisors of 351?

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 351 by each of the numbers from 1 to 351 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)

351 / 1 = 351 (the remainder is 0, so 1 is a divisor of 351)
351 / 2 = 175.5 (the remainder is 1, so 2 is not a divisor of 351)
351 / 3 = 117 (the remainder is 0, so 3 is a divisor of 351)
...
351 / 350 = 1.0028571428571 (the remainder is 1, so 350 is not a divisor of 351)
351 / 351 = 1 (the remainder is 0, so 351 is a divisor of 351)

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 351 (i.e. 18.734993995195). 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 $\frac{N}{d} = D$ )

351 / 1 = 351 (the remainder is 0, so 1 and 351 are divisors of 351)
351 / 2 = 175.5 (the remainder is 1, so 2 is not a divisor of 351)
351 / 3 = 117 (the remainder is 0, so 3 and 117 are divisors of 351)
...
351 / 17 = 20.647058823529 (the remainder is 11, so 17 is not a divisor of 351)
351 / 18 = 19.5 (the remainder is 9, so 18 is not a divisor of 351)