What are the divisors of 1227?

1, 3, 409, 1227

There is a total of 4 positive divisors.
The sum of these divisors is 1640.
The arithmetic mean is 410.

4 odd divisors

1, 3, 409, 1227

How to compute the divisors of 1227?

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 1227 by each of the numbers from 1 to 1227 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)

1227 / 1 = 1227 (the remainder is 0, so 1 is a divisor of 1227)
1227 / 2 = 613.5 (the remainder is 1, so 2 is not a divisor of 1227)
1227 / 3 = 409 (the remainder is 0, so 3 is a divisor of 1227)
...
1227 / 1226 = 1.0008156606852 (the remainder is 1, so 1226 is not a divisor of 1227)
1227 / 1227 = 1 (the remainder is 0, so 1227 is a divisor of 1227)

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 1227 (i.e. 35.028559776274). 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 $\frac{N}{d} = D$ )

1227 / 1 = 1227 (the remainder is 0, so 1 and 1227 are divisors of 1227)
1227 / 2 = 613.5 (the remainder is 1, so 2 is not a divisor of 1227)
1227 / 3 = 409 (the remainder is 0, so 3 and 409 are divisors of 1227)
...
1227 / 34 = 36.088235294118 (the remainder is 3, so 34 is not a divisor of 1227)
1227 / 35 = 35.057142857143 (the remainder is 2, so 35 is not a divisor of 1227)