What are the divisors of 1465?

1, 5, 293, 1465

There is a total of 4 positive divisors.
The sum of these divisors is 1764.
The arithmetic mean is 441.

4 odd divisors

1, 5, 293, 1465

How to compute the divisors of 1465?

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 1465 by each of the numbers from 1 to 1465 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)

1465 / 1 = 1465 (the remainder is 0, so 1 is a divisor of 1465)
1465 / 2 = 732.5 (the remainder is 1, so 2 is not a divisor of 1465)
1465 / 3 = 488.33333333333 (the remainder is 1, so 3 is not a divisor of 1465)
...
1465 / 1464 = 1.0006830601093 (the remainder is 1, so 1464 is not a divisor of 1465)
1465 / 1465 = 1 (the remainder is 0, so 1465 is a divisor of 1465)

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 1465 (i.e. 38.275318418009). 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 $\frac{N}{d} = D$ )

1465 / 1 = 1465 (the remainder is 0, so 1 and 1465 are divisors of 1465)
1465 / 2 = 732.5 (the remainder is 1, so 2 is not a divisor of 1465)
1465 / 3 = 488.33333333333 (the remainder is 1, so 3 is not a divisor of 1465)
...
1465 / 37 = 39.594594594595 (the remainder is 22, so 37 is not a divisor of 1465)
1465 / 38 = 38.552631578947 (the remainder is 21, so 38 is not a divisor of 1465)