What are the divisors of 462?

1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462

There is a total of 16 positive divisors.
The sum of these divisors is 1152.
The arithmetic mean is 72.

8 even divisors

2, 6, 14, 22, 42, 66, 154, 462

8 odd divisors

1, 3, 7, 11, 21, 33, 77, 231

How to compute the divisors of 462?

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 462 by each of the numbers from 1 to 462 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)

462 / 1 = 462 (the remainder is 0, so 1 is a divisor of 462)
462 / 2 = 231 (the remainder is 0, so 2 is a divisor of 462)
462 / 3 = 154 (the remainder is 0, so 3 is a divisor of 462)
...
462 / 461 = 1.002169197397 (the remainder is 1, so 461 is not a divisor of 462)
462 / 462 = 1 (the remainder is 0, so 462 is a divisor of 462)

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 462 (i.e. 21.494185260205). 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 $\frac{N}{d} = D$ )

462 / 1 = 462 (the remainder is 0, so 1 and 462 are divisors of 462)
462 / 2 = 231 (the remainder is 0, so 2 and 231 are divisors of 462)
462 / 3 = 154 (the remainder is 0, so 3 and 154 are divisors of 462)
...
462 / 20 = 23.1 (the remainder is 2, so 20 is not a divisor of 462)
462 / 21 = 22 (the remainder is 0, so 21 and 22 are divisors of 462)