What are the divisors of 2207?

1, 2207

There is a total of 2 positive divisors.
The sum of these divisors is 2208.
The arithmetic mean is 1104.

2 odd divisors

1, 2207

How to compute the divisors of 2207?

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 2207 by each of the numbers from 1 to 2207 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)

2207 / 1 = 2207 (the remainder is 0, so 1 is a divisor of 2207)
2207 / 2 = 1103.5 (the remainder is 1, so 2 is not a divisor of 2207)
2207 / 3 = 735.66666666667 (the remainder is 2, so 3 is not a divisor of 2207)
...
2207 / 2206 = 1.0004533091568 (the remainder is 1, so 2206 is not a divisor of 2207)
2207 / 2207 = 1 (the remainder is 0, so 2207 is a divisor of 2207)

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 2207 (i.e. 46.978718586185). 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 $\frac{N}{d} = D$ )

2207 / 1 = 2207 (the remainder is 0, so 1 and 2207 are divisors of 2207)
2207 / 2 = 1103.5 (the remainder is 1, so 2 is not a divisor of 2207)
2207 / 3 = 735.66666666667 (the remainder is 2, so 3 is not a divisor of 2207)
...
2207 / 45 = 49.044444444444 (the remainder is 2, so 45 is not a divisor of 2207)
2207 / 46 = 47.978260869565 (the remainder is 45, so 46 is not a divisor of 2207)