What are the divisors of 415?

1, 5, 83, 415

There is a total of 4 positive divisors.
The sum of these divisors is 504.
The arithmetic mean is 126.

4 odd divisors

1, 5, 83, 415

How to compute the divisors of 415?

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 415 by each of the numbers from 1 to 415 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)

415 / 1 = 415 (the remainder is 0, so 1 is a divisor of 415)
415 / 2 = 207.5 (the remainder is 1, so 2 is not a divisor of 415)
415 / 3 = 138.33333333333 (the remainder is 1, so 3 is not a divisor of 415)
...
415 / 414 = 1.0024154589372 (the remainder is 1, so 414 is not a divisor of 415)
415 / 415 = 1 (the remainder is 0, so 415 is a divisor of 415)

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 415 (i.e. 20.371548787463). 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 $\frac{N}{d} = D$ )

415 / 1 = 415 (the remainder is 0, so 1 and 415 are divisors of 415)
415 / 2 = 207.5 (the remainder is 1, so 2 is not a divisor of 415)
415 / 3 = 138.33333333333 (the remainder is 1, so 3 is not a divisor of 415)
...
415 / 19 = 21.842105263158 (the remainder is 16, so 19 is not a divisor of 415)
415 / 20 = 20.75 (the remainder is 15, so 20 is not a divisor of 415)