What are the divisors of 2201?

1, 31, 71, 2201

There is a total of 4 positive divisors.
The sum of these divisors is 2304.
The arithmetic mean is 576.

4 odd divisors

1, 31, 71, 2201

How to compute the divisors of 2201?

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 2201 by each of the numbers from 1 to 2201 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)

2201 / 1 = 2201 (the remainder is 0, so 1 is a divisor of 2201)
2201 / 2 = 1100.5 (the remainder is 1, so 2 is not a divisor of 2201)
2201 / 3 = 733.66666666667 (the remainder is 2, so 3 is not a divisor of 2201)
...
2201 / 2200 = 1.0004545454545 (the remainder is 1, so 2200 is not a divisor of 2201)
2201 / 2201 = 1 (the remainder is 0, so 2201 is a divisor of 2201)

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 2201 (i.e. 46.91481642296). 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 $\frac{N}{d} = D$ )

2201 / 1 = 2201 (the remainder is 0, so 1 and 2201 are divisors of 2201)
2201 / 2 = 1100.5 (the remainder is 1, so 2 is not a divisor of 2201)
2201 / 3 = 733.66666666667 (the remainder is 2, so 3 is not a divisor of 2201)
...
2201 / 45 = 48.911111111111 (the remainder is 41, so 45 is not a divisor of 2201)
2201 / 46 = 47.847826086957 (the remainder is 39, so 46 is not a divisor of 2201)