What are the divisors of 1724?

1, 2, 4, 431, 862, 1724

There is a total of 6 positive divisors.
The sum of these divisors is 3024.
The arithmetic mean is 504.

4 even divisors

2, 4, 862, 1724

2 odd divisors

1, 431

How to compute the divisors of 1724?

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 1724 by each of the numbers from 1 to 1724 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)

1724 / 1 = 1724 (the remainder is 0, so 1 is a divisor of 1724)
1724 / 2 = 862 (the remainder is 0, so 2 is a divisor of 1724)
1724 / 3 = 574.66666666667 (the remainder is 2, so 3 is not a divisor of 1724)
...
1724 / 1723 = 1.0005803830528 (the remainder is 1, so 1723 is not a divisor of 1724)
1724 / 1724 = 1 (the remainder is 0, so 1724 is a divisor of 1724)

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 1724 (i.e. 41.521078984053). 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 $\frac{N}{d} = D$ )

1724 / 1 = 1724 (the remainder is 0, so 1 and 1724 are divisors of 1724)
1724 / 2 = 862 (the remainder is 0, so 2 and 862 are divisors of 1724)
1724 / 3 = 574.66666666667 (the remainder is 2, so 3 is not a divisor of 1724)
...
1724 / 40 = 43.1 (the remainder is 4, so 40 is not a divisor of 1724)
1724 / 41 = 42.048780487805 (the remainder is 2, so 41 is not a divisor of 1724)