What are the divisors of 1197?

1, 3, 7, 9, 19, 21, 57, 63, 133, 171, 399, 1197

There is a total of 12 positive divisors.
The sum of these divisors is 2080.
The arithmetic mean is 173.33333333333.

12 odd divisors

1, 3, 7, 9, 19, 21, 57, 63, 133, 171, 399, 1197

How to compute the divisors of 1197?

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 1197 by each of the numbers from 1 to 1197 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)

1197 / 1 = 1197 (the remainder is 0, so 1 is a divisor of 1197)
1197 / 2 = 598.5 (the remainder is 1, so 2 is not a divisor of 1197)
1197 / 3 = 399 (the remainder is 0, so 3 is a divisor of 1197)
...
1197 / 1196 = 1.0008361204013 (the remainder is 1, so 1196 is not a divisor of 1197)
1197 / 1197 = 1 (the remainder is 0, so 1197 is a divisor of 1197)

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 1197 (i.e. 34.597687784012). 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 $\frac{N}{d} = D$ )

1197 / 1 = 1197 (the remainder is 0, so 1 and 1197 are divisors of 1197)
1197 / 2 = 598.5 (the remainder is 1, so 2 is not a divisor of 1197)
1197 / 3 = 399 (the remainder is 0, so 3 and 399 are divisors of 1197)
...
1197 / 33 = 36.272727272727 (the remainder is 9, so 33 is not a divisor of 1197)
1197 / 34 = 35.205882352941 (the remainder is 7, so 34 is not a divisor of 1197)