What are the divisors of 434?

1, 2, 7, 14, 31, 62, 217, 434

There is a total of 8 positive divisors.
The sum of these divisors is 768.
The arithmetic mean is 96.

4 even divisors

2, 14, 62, 434

4 odd divisors

1, 7, 31, 217

How to compute the divisors of 434?

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 434 by each of the numbers from 1 to 434 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)

434 / 1 = 434 (the remainder is 0, so 1 is a divisor of 434)
434 / 2 = 217 (the remainder is 0, so 2 is a divisor of 434)
434 / 3 = 144.66666666667 (the remainder is 2, so 3 is not a divisor of 434)
...
434 / 433 = 1.0023094688222 (the remainder is 1, so 433 is not a divisor of 434)
434 / 434 = 1 (the remainder is 0, so 434 is a divisor of 434)

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 434 (i.e. 20.832666656). 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 $\frac{N}{d} = D$ )

434 / 1 = 434 (the remainder is 0, so 1 and 434 are divisors of 434)
434 / 2 = 217 (the remainder is 0, so 2 and 217 are divisors of 434)
434 / 3 = 144.66666666667 (the remainder is 2, so 3 is not a divisor of 434)
...
434 / 19 = 22.842105263158 (the remainder is 16, so 19 is not a divisor of 434)
434 / 20 = 21.7 (the remainder is 14, so 20 is not a divisor of 434)