What are the divisors of 1221?

1, 3, 11, 33, 37, 111, 407, 1221

There is a total of 8 positive divisors.
The sum of these divisors is 1824.
The arithmetic mean is 228.

8 odd divisors

1, 3, 11, 33, 37, 111, 407, 1221

How to compute the divisors of 1221?

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 1221 by each of the numbers from 1 to 1221 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)

1221 / 1 = 1221 (the remainder is 0, so 1 is a divisor of 1221)
1221 / 2 = 610.5 (the remainder is 1, so 2 is not a divisor of 1221)
1221 / 3 = 407 (the remainder is 0, so 3 is a divisor of 1221)
...
1221 / 1220 = 1.0008196721311 (the remainder is 1, so 1220 is not a divisor of 1221)
1221 / 1221 = 1 (the remainder is 0, so 1221 is a divisor of 1221)

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 1221 (i.e. 34.942810419312). 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 $\frac{N}{d} = D$ )

1221 / 1 = 1221 (the remainder is 0, so 1 and 1221 are divisors of 1221)
1221 / 2 = 610.5 (the remainder is 1, so 2 is not a divisor of 1221)
1221 / 3 = 407 (the remainder is 0, so 3 and 407 are divisors of 1221)
...
1221 / 33 = 37 (the remainder is 0, so 33 and 37 are divisors of 1221)
1221 / 34 = 35.911764705882 (the remainder is 31, so 34 is not a divisor of 1221)