What are the divisors of 1722?

1, 2, 3, 6, 7, 14, 21, 41, 42, 82, 123, 246, 287, 574, 861, 1722

There is a total of 16 positive divisors.
The sum of these divisors is 4032.
The arithmetic mean is 252.

8 even divisors

2, 6, 14, 42, 82, 246, 574, 1722

8 odd divisors

1, 3, 7, 21, 41, 123, 287, 861

How to compute the divisors of 1722?

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 1722 by each of the numbers from 1 to 1722 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)

1722 / 1 = 1722 (the remainder is 0, so 1 is a divisor of 1722)
1722 / 2 = 861 (the remainder is 0, so 2 is a divisor of 1722)
1722 / 3 = 574 (the remainder is 0, so 3 is a divisor of 1722)
...
1722 / 1721 = 1.0005810575247 (the remainder is 1, so 1721 is not a divisor of 1722)
1722 / 1722 = 1 (the remainder is 0, so 1722 is a divisor of 1722)

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 1722 (i.e. 41.496987842493). 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 $\frac{N}{d} = D$ )

1722 / 1 = 1722 (the remainder is 0, so 1 and 1722 are divisors of 1722)
1722 / 2 = 861 (the remainder is 0, so 2 and 861 are divisors of 1722)
1722 / 3 = 574 (the remainder is 0, so 3 and 574 are divisors of 1722)
...
1722 / 40 = 43.05 (the remainder is 2, so 40 is not a divisor of 1722)
1722 / 41 = 42 (the remainder is 0, so 41 and 42 are divisors of 1722)