What are the divisors of 1415?

1, 5, 283, 1415

There is a total of 4 positive divisors.
The sum of these divisors is 1704.
The arithmetic mean is 426.

4 odd divisors

1, 5, 283, 1415

How to compute the divisors of 1415?

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 1415 by each of the numbers from 1 to 1415 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)

1415 / 1 = 1415 (the remainder is 0, so 1 is a divisor of 1415)
1415 / 2 = 707.5 (the remainder is 1, so 2 is not a divisor of 1415)
1415 / 3 = 471.66666666667 (the remainder is 2, so 3 is not a divisor of 1415)
...
1415 / 1414 = 1.0007072135785 (the remainder is 1, so 1414 is not a divisor of 1415)
1415 / 1415 = 1 (the remainder is 0, so 1415 is a divisor of 1415)

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 1415 (i.e. 37.616485747608). 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 $\frac{N}{d} = D$ )

1415 / 1 = 1415 (the remainder is 0, so 1 and 1415 are divisors of 1415)
1415 / 2 = 707.5 (the remainder is 1, so 2 is not a divisor of 1415)
1415 / 3 = 471.66666666667 (the remainder is 2, so 3 is not a divisor of 1415)
...
1415 / 36 = 39.305555555556 (the remainder is 11, so 36 is not a divisor of 1415)
1415 / 37 = 38.243243243243 (the remainder is 9, so 37 is not a divisor of 1415)