What are the divisors of 3197?

1, 23, 139, 3197

There is a total of 4 positive divisors.
The sum of these divisors is 3360.
The arithmetic mean is 840.

4 odd divisors

1, 23, 139, 3197

How to compute the divisors of 3197?

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 3197 by each of the numbers from 1 to 3197 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)

3197 / 1 = 3197 (the remainder is 0, so 1 is a divisor of 3197)
3197 / 2 = 1598.5 (the remainder is 1, so 2 is not a divisor of 3197)
3197 / 3 = 1065.6666666667 (the remainder is 2, so 3 is not a divisor of 3197)
...
3197 / 3196 = 1.0003128911139 (the remainder is 1, so 3196 is not a divisor of 3197)
3197 / 3197 = 1 (the remainder is 0, so 3197 is a divisor of 3197)

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 3197 (i.e. 56.542019772909). 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 $\frac{N}{d} = D$ )

3197 / 1 = 3197 (the remainder is 0, so 1 and 3197 are divisors of 3197)
3197 / 2 = 1598.5 (the remainder is 1, so 2 is not a divisor of 3197)
3197 / 3 = 1065.6666666667 (the remainder is 2, so 3 is not a divisor of 3197)
...
3197 / 55 = 58.127272727273 (the remainder is 7, so 55 is not a divisor of 3197)
3197 / 56 = 57.089285714286 (the remainder is 5, so 56 is not a divisor of 3197)