What are the divisors of 1345?

1, 5, 269, 1345

There is a total of 4 positive divisors.
The sum of these divisors is 1620.
The arithmetic mean is 405.

4 odd divisors

1, 5, 269, 1345

How to compute the divisors of 1345?

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 1345 by each of the numbers from 1 to 1345 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)

1345 / 1 = 1345 (the remainder is 0, so 1 is a divisor of 1345)
1345 / 2 = 672.5 (the remainder is 1, so 2 is not a divisor of 1345)
1345 / 3 = 448.33333333333 (the remainder is 1, so 3 is not a divisor of 1345)
...
1345 / 1344 = 1.000744047619 (the remainder is 1, so 1344 is not a divisor of 1345)
1345 / 1345 = 1 (the remainder is 0, so 1345 is a divisor of 1345)

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 1345 (i.e. 36.674241641784). 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 $\frac{N}{d} = D$ )

1345 / 1 = 1345 (the remainder is 0, so 1 and 1345 are divisors of 1345)
1345 / 2 = 672.5 (the remainder is 1, so 2 is not a divisor of 1345)
1345 / 3 = 448.33333333333 (the remainder is 1, so 3 is not a divisor of 1345)
...
1345 / 35 = 38.428571428571 (the remainder is 15, so 35 is not a divisor of 1345)
1345 / 36 = 37.361111111111 (the remainder is 13, so 36 is not a divisor of 1345)