What are the divisors of 1461?

1, 3, 487, 1461

There is a total of 4 positive divisors.
The sum of these divisors is 1952.
The arithmetic mean is 488.

4 odd divisors

1, 3, 487, 1461

How to compute the divisors of 1461?

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 1461 by each of the numbers from 1 to 1461 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)

1461 / 1 = 1461 (the remainder is 0, so 1 is a divisor of 1461)
1461 / 2 = 730.5 (the remainder is 1, so 2 is not a divisor of 1461)
1461 / 3 = 487 (the remainder is 0, so 3 is a divisor of 1461)
...
1461 / 1460 = 1.0006849315068 (the remainder is 1, so 1460 is not a divisor of 1461)
1461 / 1461 = 1 (the remainder is 0, so 1461 is a divisor of 1461)

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 1461 (i.e. 38.223029707233). 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 $\frac{N}{d} = D$ )

1461 / 1 = 1461 (the remainder is 0, so 1 and 1461 are divisors of 1461)
1461 / 2 = 730.5 (the remainder is 1, so 2 is not a divisor of 1461)
1461 / 3 = 487 (the remainder is 0, so 3 and 487 are divisors of 1461)
...
1461 / 37 = 39.486486486486 (the remainder is 18, so 37 is not a divisor of 1461)
1461 / 38 = 38.447368421053 (the remainder is 17, so 38 is not a divisor of 1461)