What are the divisors of 1497?

1, 3, 499, 1497

There is a total of 4 positive divisors.
The sum of these divisors is 2000.
The arithmetic mean is 500.

4 odd divisors

1, 3, 499, 1497

How to compute the divisors of 1497?

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 1497 by each of the numbers from 1 to 1497 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)

1497 / 1 = 1497 (the remainder is 0, so 1 is a divisor of 1497)
1497 / 2 = 748.5 (the remainder is 1, so 2 is not a divisor of 1497)
1497 / 3 = 499 (the remainder is 0, so 3 is a divisor of 1497)
...
1497 / 1496 = 1.0006684491979 (the remainder is 1, so 1496 is not a divisor of 1497)
1497 / 1497 = 1 (the remainder is 0, so 1497 is a divisor of 1497)

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 1497 (i.e. 38.691084244306). 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 $\frac{N}{d} = D$ )

1497 / 1 = 1497 (the remainder is 0, so 1 and 1497 are divisors of 1497)
1497 / 2 = 748.5 (the remainder is 1, so 2 is not a divisor of 1497)
1497 / 3 = 499 (the remainder is 0, so 3 and 499 are divisors of 1497)
...
1497 / 37 = 40.459459459459 (the remainder is 17, so 37 is not a divisor of 1497)
1497 / 38 = 39.394736842105 (the remainder is 15, so 38 is not a divisor of 1497)