What are the divisors of 402?

1, 2, 3, 6, 67, 134, 201, 402

There is a total of 8 positive divisors.
The sum of these divisors is 816.
The arithmetic mean is 102.

4 even divisors

2, 6, 134, 402

4 odd divisors

1, 3, 67, 201

How to compute the divisors of 402?

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 402 by each of the numbers from 1 to 402 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)

402 / 1 = 402 (the remainder is 0, so 1 is a divisor of 402)
402 / 2 = 201 (the remainder is 0, so 2 is a divisor of 402)
402 / 3 = 134 (the remainder is 0, so 3 is a divisor of 402)
...
402 / 401 = 1.002493765586 (the remainder is 1, so 401 is not a divisor of 402)
402 / 402 = 1 (the remainder is 0, so 402 is a divisor of 402)

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 402 (i.e. 20.049937655763). 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 $\frac{N}{d} = D$ )

402 / 1 = 402 (the remainder is 0, so 1 and 402 are divisors of 402)
402 / 2 = 201 (the remainder is 0, so 2 and 201 are divisors of 402)
402 / 3 = 134 (the remainder is 0, so 3 and 134 are divisors of 402)
...
402 / 19 = 21.157894736842 (the remainder is 3, so 19 is not a divisor of 402)
402 / 20 = 20.1 (the remainder is 2, so 20 is not a divisor of 402)