What are the divisors of 353?

1, 353

There is a total of 2 positive divisors.
The sum of these divisors is 354.
The arithmetic mean is 177.

2 odd divisors

1, 353

How to compute the divisors of 353?

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 353 by each of the numbers from 1 to 353 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)

353 / 1 = 353 (the remainder is 0, so 1 is a divisor of 353)
353 / 2 = 176.5 (the remainder is 1, so 2 is not a divisor of 353)
353 / 3 = 117.66666666667 (the remainder is 2, so 3 is not a divisor of 353)
...
353 / 352 = 1.0028409090909 (the remainder is 1, so 352 is not a divisor of 353)
353 / 353 = 1 (the remainder is 0, so 353 is a divisor of 353)

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 353 (i.e. 18.788294228056). 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 $\frac{N}{d} = D$ )

353 / 1 = 353 (the remainder is 0, so 1 and 353 are divisors of 353)
353 / 2 = 176.5 (the remainder is 1, so 2 is not a divisor of 353)
353 / 3 = 117.66666666667 (the remainder is 2, so 3 is not a divisor of 353)
...
353 / 17 = 20.764705882353 (the remainder is 13, so 17 is not a divisor of 353)
353 / 18 = 19.611111111111 (the remainder is 11, so 18 is not a divisor of 353)