What are the divisors of 665?

1, 5, 7, 19, 35, 95, 133, 665

There is a total of 8 positive divisors.
The sum of these divisors is 960.
The arithmetic mean is 120.

8 odd divisors

1, 5, 7, 19, 35, 95, 133, 665

How to compute the divisors of 665?

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 665 by each of the numbers from 1 to 665 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)

665 / 1 = 665 (the remainder is 0, so 1 is a divisor of 665)
665 / 2 = 332.5 (the remainder is 1, so 2 is not a divisor of 665)
665 / 3 = 221.66666666667 (the remainder is 2, so 3 is not a divisor of 665)
...
665 / 664 = 1.0015060240964 (the remainder is 1, so 664 is not a divisor of 665)
665 / 665 = 1 (the remainder is 0, so 665 is a divisor of 665)

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 665 (i.e. 25.787593916455). 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 $\frac{N}{d} = D$ )

665 / 1 = 665 (the remainder is 0, so 1 and 665 are divisors of 665)
665 / 2 = 332.5 (the remainder is 1, so 2 is not a divisor of 665)
665 / 3 = 221.66666666667 (the remainder is 2, so 3 is not a divisor of 665)
...
665 / 24 = 27.708333333333 (the remainder is 17, so 24 is not a divisor of 665)
665 / 25 = 26.6 (the remainder is 15, so 25 is not a divisor of 665)