What are the divisors of 414?

1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414

There is a total of 12 positive divisors.
The sum of these divisors is 936.
The arithmetic mean is 78.

6 even divisors

2, 6, 18, 46, 138, 414

6 odd divisors

1, 3, 9, 23, 69, 207

How to compute the divisors of 414?

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 414 by each of the numbers from 1 to 414 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)

414 / 1 = 414 (the remainder is 0, so 1 is a divisor of 414)
414 / 2 = 207 (the remainder is 0, so 2 is a divisor of 414)
414 / 3 = 138 (the remainder is 0, so 3 is a divisor of 414)
...
414 / 413 = 1.0024213075061 (the remainder is 1, so 413 is not a divisor of 414)
414 / 414 = 1 (the remainder is 0, so 414 is a divisor of 414)

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 414 (i.e. 20.346989949376). 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 $\frac{N}{d} = D$ )

414 / 1 = 414 (the remainder is 0, so 1 and 414 are divisors of 414)
414 / 2 = 207 (the remainder is 0, so 2 and 207 are divisors of 414)
414 / 3 = 138 (the remainder is 0, so 3 and 138 are divisors of 414)
...
414 / 19 = 21.789473684211 (the remainder is 15, so 19 is not a divisor of 414)
414 / 20 = 20.7 (the remainder is 14, so 20 is not a divisor of 414)