What are the divisors of 4197?

1, 3, 1399, 4197

There is a total of 4 positive divisors.
The sum of these divisors is 5600.
The arithmetic mean is 1400.

4 odd divisors

1, 3, 1399, 4197

How to compute the divisors of 4197?

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 4197 by each of the numbers from 1 to 4197 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)

4197 / 1 = 4197 (the remainder is 0, so 1 is a divisor of 4197)
4197 / 2 = 2098.5 (the remainder is 1, so 2 is not a divisor of 4197)
4197 / 3 = 1399 (the remainder is 0, so 3 is a divisor of 4197)
...
4197 / 4196 = 1.0002383222116 (the remainder is 1, so 4196 is not a divisor of 4197)
4197 / 4197 = 1 (the remainder is 0, so 4197 is a divisor of 4197)

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 4197 (i.e. 64.784257346982). 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 $\frac{N}{d} = D$ )

4197 / 1 = 4197 (the remainder is 0, so 1 and 4197 are divisors of 4197)
4197 / 2 = 2098.5 (the remainder is 1, so 2 is not a divisor of 4197)
4197 / 3 = 1399 (the remainder is 0, so 3 and 1399 are divisors of 4197)
...
4197 / 63 = 66.619047619048 (the remainder is 39, so 63 is not a divisor of 4197)
4197 / 64 = 65.578125 (the remainder is 37, so 64 is not a divisor of 4197)