What are the divisors of 357?

1, 3, 7, 17, 21, 51, 119, 357

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, 7, 17, 21, 51, 119, 357

How to compute the divisors of 357?

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 357 by each of the numbers from 1 to 357 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)

357 / 1 = 357 (the remainder is 0, so 1 is a divisor of 357)
357 / 2 = 178.5 (the remainder is 1, so 2 is not a divisor of 357)
357 / 3 = 119 (the remainder is 0, so 3 is a divisor of 357)
...
357 / 356 = 1.002808988764 (the remainder is 1, so 356 is not a divisor of 357)
357 / 357 = 1 (the remainder is 0, so 357 is a divisor of 357)

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 357 (i.e. 18.894443627691). 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 $\frac{N}{d} = D$ )

357 / 1 = 357 (the remainder is 0, so 1 and 357 are divisors of 357)
357 / 2 = 178.5 (the remainder is 1, so 2 is not a divisor of 357)
357 / 3 = 119 (the remainder is 0, so 3 and 119 are divisors of 357)
...
357 / 17 = 21 (the remainder is 0, so 17 and 21 are divisors of 357)
357 / 18 = 19.833333333333 (the remainder is 15, so 18 is not a divisor of 357)