What are the divisors of 1491?

1, 3, 7, 21, 71, 213, 497, 1491

There is a total of 8 positive divisors.
The sum of these divisors is 2304.
The arithmetic mean is 288.

8 odd divisors

1, 3, 7, 21, 71, 213, 497, 1491

How to compute the divisors of 1491?

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 1491 by each of the numbers from 1 to 1491 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)

1491 / 1 = 1491 (the remainder is 0, so 1 is a divisor of 1491)
1491 / 2 = 745.5 (the remainder is 1, so 2 is not a divisor of 1491)
1491 / 3 = 497 (the remainder is 0, so 3 is a divisor of 1491)
...
1491 / 1490 = 1.0006711409396 (the remainder is 1, so 1490 is not a divisor of 1491)
1491 / 1491 = 1 (the remainder is 0, so 1491 is a divisor of 1491)

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 1491 (i.e. 38.613469152616). 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 $\frac{N}{d} = D$ )

1491 / 1 = 1491 (the remainder is 0, so 1 and 1491 are divisors of 1491)
1491 / 2 = 745.5 (the remainder is 1, so 2 is not a divisor of 1491)
1491 / 3 = 497 (the remainder is 0, so 3 and 497 are divisors of 1491)
...
1491 / 37 = 40.297297297297 (the remainder is 11, so 37 is not a divisor of 1491)
1491 / 38 = 39.236842105263 (the remainder is 9, so 38 is not a divisor of 1491)