What are the divisors of 658?

1, 2, 7, 14, 47, 94, 329, 658

There is a total of 8 positive divisors.
The sum of these divisors is 1152.
The arithmetic mean is 144.

4 even divisors

2, 14, 94, 658

4 odd divisors

1, 7, 47, 329

How to compute the divisors of 658?

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 658 by each of the numbers from 1 to 658 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)

658 / 1 = 658 (the remainder is 0, so 1 is a divisor of 658)
658 / 2 = 329 (the remainder is 0, so 2 is a divisor of 658)
658 / 3 = 219.33333333333 (the remainder is 1, so 3 is not a divisor of 658)
...
658 / 657 = 1.0015220700152 (the remainder is 1, so 657 is not a divisor of 658)
658 / 658 = 1 (the remainder is 0, so 658 is a divisor of 658)

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 658 (i.e. 25.651510676761). 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 $\frac{N}{d} = D$ )

658 / 1 = 658 (the remainder is 0, so 1 and 658 are divisors of 658)
658 / 2 = 329 (the remainder is 0, so 2 and 329 are divisors of 658)
658 / 3 = 219.33333333333 (the remainder is 1, so 3 is not a divisor of 658)
...
658 / 24 = 27.416666666667 (the remainder is 10, so 24 is not a divisor of 658)
658 / 25 = 26.32 (the remainder is 8, so 25 is not a divisor of 658)