What are the divisors of 2353?

1, 13, 181, 2353

There is a total of 4 positive divisors.
The sum of these divisors is 2548.
The arithmetic mean is 637.

4 odd divisors

1, 13, 181, 2353

How to compute the divisors of 2353?

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 2353 by each of the numbers from 1 to 2353 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)

2353 / 1 = 2353 (the remainder is 0, so 1 is a divisor of 2353)
2353 / 2 = 1176.5 (the remainder is 1, so 2 is not a divisor of 2353)
2353 / 3 = 784.33333333333 (the remainder is 1, so 3 is not a divisor of 2353)
...
2353 / 2352 = 1.000425170068 (the remainder is 1, so 2352 is not a divisor of 2353)
2353 / 2353 = 1 (the remainder is 0, so 2353 is a divisor of 2353)

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 2353 (i.e. 48.50773134254). 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 $\frac{N}{d} = D$ )

2353 / 1 = 2353 (the remainder is 0, so 1 and 2353 are divisors of 2353)
2353 / 2 = 1176.5 (the remainder is 1, so 2 is not a divisor of 2353)
2353 / 3 = 784.33333333333 (the remainder is 1, so 3 is not a divisor of 2353)
...
2353 / 47 = 50.063829787234 (the remainder is 3, so 47 is not a divisor of 2353)
2353 / 48 = 49.020833333333 (the remainder is 1, so 48 is not a divisor of 2353)