What are the divisors of 429?

1, 3, 11, 13, 33, 39, 143, 429

There is a total of 8 positive divisors.
The sum of these divisors is 672.
The arithmetic mean is 84.

8 odd divisors

1, 3, 11, 13, 33, 39, 143, 429

How to compute the divisors of 429?

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 429 by each of the numbers from 1 to 429 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)

429 / 1 = 429 (the remainder is 0, so 1 is a divisor of 429)
429 / 2 = 214.5 (the remainder is 1, so 2 is not a divisor of 429)
429 / 3 = 143 (the remainder is 0, so 3 is a divisor of 429)
...
429 / 428 = 1.0023364485981 (the remainder is 1, so 428 is not a divisor of 429)
429 / 429 = 1 (the remainder is 0, so 429 is a divisor of 429)

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 429 (i.e. 20.712315177208). 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 $\frac{N}{d} = D$ )

429 / 1 = 429 (the remainder is 0, so 1 and 429 are divisors of 429)
429 / 2 = 214.5 (the remainder is 1, so 2 is not a divisor of 429)
429 / 3 = 143 (the remainder is 0, so 3 and 143 are divisors of 429)
...
429 / 19 = 22.578947368421 (the remainder is 11, so 19 is not a divisor of 429)
429 / 20 = 21.45 (the remainder is 9, so 20 is not a divisor of 429)