What are the divisors of 1572?

1, 2, 3, 4, 6, 12, 131, 262, 393, 524, 786, 1572

There is a total of 12 positive divisors.
The sum of these divisors is 3696.
The arithmetic mean is 308.

8 even divisors

2, 4, 6, 12, 262, 524, 786, 1572

4 odd divisors

1, 3, 131, 393

How to compute the divisors of 1572?

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 1572 by each of the numbers from 1 to 1572 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)

1572 / 1 = 1572 (the remainder is 0, so 1 is a divisor of 1572)
1572 / 2 = 786 (the remainder is 0, so 2 is a divisor of 1572)
1572 / 3 = 524 (the remainder is 0, so 3 is a divisor of 1572)
...
1572 / 1571 = 1.0006365372374 (the remainder is 1, so 1571 is not a divisor of 1572)
1572 / 1572 = 1 (the remainder is 0, so 1572 is a divisor of 1572)

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 1572 (i.e. 39.648455203198). 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 $\frac{N}{d} = D$ )

1572 / 1 = 1572 (the remainder is 0, so 1 and 1572 are divisors of 1572)
1572 / 2 = 786 (the remainder is 0, so 2 and 786 are divisors of 1572)
1572 / 3 = 524 (the remainder is 0, so 3 and 524 are divisors of 1572)
...
1572 / 38 = 41.368421052632 (the remainder is 14, so 38 is not a divisor of 1572)
1572 / 39 = 40.307692307692 (the remainder is 12, so 39 is not a divisor of 1572)