What are the divisors of 1405?

1, 5, 281, 1405

There is a total of 4 positive divisors.
The sum of these divisors is 1692.
The arithmetic mean is 423.

4 odd divisors

1, 5, 281, 1405

How to compute the divisors of 1405?

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 1405 by each of the numbers from 1 to 1405 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)

1405 / 1 = 1405 (the remainder is 0, so 1 is a divisor of 1405)
1405 / 2 = 702.5 (the remainder is 1, so 2 is not a divisor of 1405)
1405 / 3 = 468.33333333333 (the remainder is 1, so 3 is not a divisor of 1405)
...
1405 / 1404 = 1.0007122507123 (the remainder is 1, so 1404 is not a divisor of 1405)
1405 / 1405 = 1 (the remainder is 0, so 1405 is a divisor of 1405)

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 1405 (i.e. 37.483329627983). 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 $\frac{N}{d} = D$ )

1405 / 1 = 1405 (the remainder is 0, so 1 and 1405 are divisors of 1405)
1405 / 2 = 702.5 (the remainder is 1, so 2 is not a divisor of 1405)
1405 / 3 = 468.33333333333 (the remainder is 1, so 3 is not a divisor of 1405)
...
1405 / 36 = 39.027777777778 (the remainder is 1, so 36 is not a divisor of 1405)
1405 / 37 = 37.972972972973 (the remainder is 36, so 37 is not a divisor of 1405)