What are the divisors of 1561?

1, 7, 223, 1561

There is a total of 4 positive divisors.
The sum of these divisors is 1792.
The arithmetic mean is 448.

4 odd divisors

1, 7, 223, 1561

How to compute the divisors of 1561?

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 1561 by each of the numbers from 1 to 1561 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)

1561 / 1 = 1561 (the remainder is 0, so 1 is a divisor of 1561)
1561 / 2 = 780.5 (the remainder is 1, so 2 is not a divisor of 1561)
1561 / 3 = 520.33333333333 (the remainder is 1, so 3 is not a divisor of 1561)
...
1561 / 1560 = 1.000641025641 (the remainder is 1, so 1560 is not a divisor of 1561)
1561 / 1561 = 1 (the remainder is 0, so 1561 is a divisor of 1561)

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 1561 (i.e. 39.509492530277). 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 $\frac{N}{d} = D$ )

1561 / 1 = 1561 (the remainder is 0, so 1 and 1561 are divisors of 1561)
1561 / 2 = 780.5 (the remainder is 1, so 2 is not a divisor of 1561)
1561 / 3 = 520.33333333333 (the remainder is 1, so 3 is not a divisor of 1561)
...
1561 / 38 = 41.078947368421 (the remainder is 3, so 38 is not a divisor of 1561)
1561 / 39 = 40.025641025641 (the remainder is 1, so 39 is not a divisor of 1561)