What are the divisors of 2197?

1, 13, 169, 2197

There is a total of 4 positive divisors.
The sum of these divisors is 2380.
The arithmetic mean is 595.

4 odd divisors

1, 13, 169, 2197

How to compute the divisors of 2197?

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 2197 by each of the numbers from 1 to 2197 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)

2197 / 1 = 2197 (the remainder is 0, so 1 is a divisor of 2197)
2197 / 2 = 1098.5 (the remainder is 1, so 2 is not a divisor of 2197)
2197 / 3 = 732.33333333333 (the remainder is 1, so 3 is not a divisor of 2197)
...
2197 / 2196 = 1.0004553734062 (the remainder is 1, so 2196 is not a divisor of 2197)
2197 / 2197 = 1 (the remainder is 0, so 2197 is a divisor of 2197)

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 2197 (i.e. 46.872166581032). 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 $\frac{N}{d} = D$ )

2197 / 1 = 2197 (the remainder is 0, so 1 and 2197 are divisors of 2197)
2197 / 2 = 1098.5 (the remainder is 1, so 2 is not a divisor of 2197)
2197 / 3 = 732.33333333333 (the remainder is 1, so 3 is not a divisor of 2197)
...
2197 / 45 = 48.822222222222 (the remainder is 37, so 45 is not a divisor of 2197)
2197 / 46 = 47.760869565217 (the remainder is 35, so 46 is not a divisor of 2197)