What are the divisors of 1677?

1, 3, 13, 39, 43, 129, 559, 1677

There is a total of 8 positive divisors.
The sum of these divisors is 2464.
The arithmetic mean is 308.

8 odd divisors

1, 3, 13, 39, 43, 129, 559, 1677

How to compute the divisors of 1677?

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 1677 by each of the numbers from 1 to 1677 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)

1677 / 1 = 1677 (the remainder is 0, so 1 is a divisor of 1677)
1677 / 2 = 838.5 (the remainder is 1, so 2 is not a divisor of 1677)
1677 / 3 = 559 (the remainder is 0, so 3 is a divisor of 1677)
...
1677 / 1676 = 1.0005966587112 (the remainder is 1, so 1676 is not a divisor of 1677)
1677 / 1677 = 1 (the remainder is 0, so 1677 is a divisor of 1677)

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 1677 (i.e. 40.951190458887). 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 $\frac{N}{d} = D$ )

1677 / 1 = 1677 (the remainder is 0, so 1 and 1677 are divisors of 1677)
1677 / 2 = 838.5 (the remainder is 1, so 2 is not a divisor of 1677)
1677 / 3 = 559 (the remainder is 0, so 3 and 559 are divisors of 1677)
...
1677 / 39 = 43 (the remainder is 0, so 39 and 43 are divisors of 1677)
1677 / 40 = 41.925 (the remainder is 37, so 40 is not a divisor of 1677)