What are the divisors of 649?

1, 11, 59, 649

There is a total of 4 positive divisors.
The sum of these divisors is 720.
The arithmetic mean is 180.

4 odd divisors

1, 11, 59, 649

How to compute the divisors of 649?

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 649 by each of the numbers from 1 to 649 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)

649 / 1 = 649 (the remainder is 0, so 1 is a divisor of 649)
649 / 2 = 324.5 (the remainder is 1, so 2 is not a divisor of 649)
649 / 3 = 216.33333333333 (the remainder is 1, so 3 is not a divisor of 649)
...
649 / 648 = 1.0015432098765 (the remainder is 1, so 648 is not a divisor of 649)
649 / 649 = 1 (the remainder is 0, so 649 is a divisor of 649)

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 649 (i.e. 25.475478405714). 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 $\frac{N}{d} = D$ )

649 / 1 = 649 (the remainder is 0, so 1 and 649 are divisors of 649)
649 / 2 = 324.5 (the remainder is 1, so 2 is not a divisor of 649)
649 / 3 = 216.33333333333 (the remainder is 1, so 3 is not a divisor of 649)
...
649 / 24 = 27.041666666667 (the remainder is 1, so 24 is not a divisor of 649)
649 / 25 = 25.96 (the remainder is 24, so 25 is not a divisor of 649)