What are the divisors of 2345?

1, 5, 7, 35, 67, 335, 469, 2345

There is a total of 8 positive divisors.
The sum of these divisors is 3264.
The arithmetic mean is 408.

8 odd divisors

1, 5, 7, 35, 67, 335, 469, 2345

How to compute the divisors of 2345?

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 2345 by each of the numbers from 1 to 2345 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)

2345 / 1 = 2345 (the remainder is 0, so 1 is a divisor of 2345)
2345 / 2 = 1172.5 (the remainder is 1, so 2 is not a divisor of 2345)
2345 / 3 = 781.66666666667 (the remainder is 2, so 3 is not a divisor of 2345)
...
2345 / 2344 = 1.0004266211604 (the remainder is 1, so 2344 is not a divisor of 2345)
2345 / 2345 = 1 (the remainder is 0, so 2345 is a divisor of 2345)

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 2345 (i.e. 48.425200051213). 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 $\frac{N}{d} = D$ )

2345 / 1 = 2345 (the remainder is 0, so 1 and 2345 are divisors of 2345)
2345 / 2 = 1172.5 (the remainder is 1, so 2 is not a divisor of 2345)
2345 / 3 = 781.66666666667 (the remainder is 2, so 3 is not a divisor of 2345)
...
2345 / 47 = 49.893617021277 (the remainder is 42, so 47 is not a divisor of 2345)
2345 / 48 = 48.854166666667 (the remainder is 41, so 48 is not a divisor of 2345)