What are the divisors of 1673?

1, 7, 239, 1673

There is a total of 4 positive divisors.
The sum of these divisors is 1920.
The arithmetic mean is 480.

4 odd divisors

1, 7, 239, 1673

How to compute the divisors of 1673?

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 1673 by each of the numbers from 1 to 1673 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)

1673 / 1 = 1673 (the remainder is 0, so 1 is a divisor of 1673)
1673 / 2 = 836.5 (the remainder is 1, so 2 is not a divisor of 1673)
1673 / 3 = 557.66666666667 (the remainder is 2, so 3 is not a divisor of 1673)
...
1673 / 1672 = 1.0005980861244 (the remainder is 1, so 1672 is not a divisor of 1673)
1673 / 1673 = 1 (the remainder is 0, so 1673 is a divisor of 1673)

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 1673 (i.e. 40.902322672435). 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 $\frac{N}{d} = D$ )

1673 / 1 = 1673 (the remainder is 0, so 1 and 1673 are divisors of 1673)
1673 / 2 = 836.5 (the remainder is 1, so 2 is not a divisor of 1673)
1673 / 3 = 557.66666666667 (the remainder is 2, so 3 is not a divisor of 1673)
...
1673 / 39 = 42.897435897436 (the remainder is 35, so 39 is not a divisor of 1673)
1673 / 40 = 41.825 (the remainder is 33, so 40 is not a divisor of 1673)