What are the divisors of 1441?

1, 11, 131, 1441

There is a total of 4 positive divisors.
The sum of these divisors is 1584.
The arithmetic mean is 396.

4 odd divisors

1, 11, 131, 1441

How to compute the divisors of 1441?

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 1441 by each of the numbers from 1 to 1441 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)

1441 / 1 = 1441 (the remainder is 0, so 1 is a divisor of 1441)
1441 / 2 = 720.5 (the remainder is 1, so 2 is not a divisor of 1441)
1441 / 3 = 480.33333333333 (the remainder is 1, so 3 is not a divisor of 1441)
...
1441 / 1440 = 1.0006944444444 (the remainder is 1, so 1440 is not a divisor of 1441)
1441 / 1441 = 1 (the remainder is 0, so 1441 is a divisor of 1441)

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 1441 (i.e. 37.960505792205). 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 $\frac{N}{d} = D$ )

1441 / 1 = 1441 (the remainder is 0, so 1 and 1441 are divisors of 1441)
1441 / 2 = 720.5 (the remainder is 1, so 2 is not a divisor of 1441)
1441 / 3 = 480.33333333333 (the remainder is 1, so 3 is not a divisor of 1441)
...
1441 / 36 = 40.027777777778 (the remainder is 1, so 36 is not a divisor of 1441)
1441 / 37 = 38.945945945946 (the remainder is 35, so 37 is not a divisor of 1441)