What are the divisors of 677?

1, 677

There is a total of 2 positive divisors.
The sum of these divisors is 678.
The arithmetic mean is 339.

2 odd divisors

1, 677

How to compute the divisors of 677?

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 677 by each of the numbers from 1 to 677 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)

677 / 1 = 677 (the remainder is 0, so 1 is a divisor of 677)
677 / 2 = 338.5 (the remainder is 1, so 2 is not a divisor of 677)
677 / 3 = 225.66666666667 (the remainder is 2, so 3 is not a divisor of 677)
...
677 / 676 = 1.0014792899408 (the remainder is 1, so 676 is not a divisor of 677)
677 / 677 = 1 (the remainder is 0, so 677 is a divisor of 677)

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 677 (i.e. 26.019223662515). 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 $\frac{N}{d} = D$ )

677 / 1 = 677 (the remainder is 0, so 1 and 677 are divisors of 677)
677 / 2 = 338.5 (the remainder is 1, so 2 is not a divisor of 677)
677 / 3 = 225.66666666667 (the remainder is 2, so 3 is not a divisor of 677)
...
677 / 25 = 27.08 (the remainder is 2, so 25 is not a divisor of 677)
677 / 26 = 26.038461538462 (the remainder is 1, so 26 is not a divisor of 677)