What are the divisors of 653?

1, 653

There is a total of 2 positive divisors.
The sum of these divisors is 654.
The arithmetic mean is 327.

2 odd divisors

1, 653

How to compute the divisors of 653?

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 653 by each of the numbers from 1 to 653 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)

653 / 1 = 653 (the remainder is 0, so 1 is a divisor of 653)
653 / 2 = 326.5 (the remainder is 1, so 2 is not a divisor of 653)
653 / 3 = 217.66666666667 (the remainder is 2, so 3 is not a divisor of 653)
...
653 / 652 = 1.0015337423313 (the remainder is 1, so 652 is not a divisor of 653)
653 / 653 = 1 (the remainder is 0, so 653 is a divisor of 653)

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 653 (i.e. 25.553864678361). 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 $\frac{N}{d} = D$ )

653 / 1 = 653 (the remainder is 0, so 1 and 653 are divisors of 653)
653 / 2 = 326.5 (the remainder is 1, so 2 is not a divisor of 653)
653 / 3 = 217.66666666667 (the remainder is 2, so 3 is not a divisor of 653)
...
653 / 24 = 27.208333333333 (the remainder is 5, so 24 is not a divisor of 653)
653 / 25 = 26.12 (the remainder is 3, so 25 is not a divisor of 653)