What are the divisors of 2151?

1, 3, 9, 239, 717, 2151

There is a total of 6 positive divisors.
The sum of these divisors is 3120.
The arithmetic mean is 520.

6 odd divisors

1, 3, 9, 239, 717, 2151

How to compute the divisors of 2151?

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 2151 by each of the numbers from 1 to 2151 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)

2151 / 1 = 2151 (the remainder is 0, so 1 is a divisor of 2151)
2151 / 2 = 1075.5 (the remainder is 1, so 2 is not a divisor of 2151)
2151 / 3 = 717 (the remainder is 0, so 3 is a divisor of 2151)
...
2151 / 2150 = 1.0004651162791 (the remainder is 1, so 2150 is not a divisor of 2151)
2151 / 2151 = 1 (the remainder is 0, so 2151 is a divisor of 2151)

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 2151 (i.e. 46.378874501221). 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 $\frac{N}{d} = D$ )

2151 / 1 = 2151 (the remainder is 0, so 1 and 2151 are divisors of 2151)
2151 / 2 = 1075.5 (the remainder is 1, so 2 is not a divisor of 2151)
2151 / 3 = 717 (the remainder is 0, so 3 and 717 are divisors of 2151)
...
2151 / 45 = 47.8 (the remainder is 36, so 45 is not a divisor of 2151)
2151 / 46 = 46.760869565217 (the remainder is 35, so 46 is not a divisor of 2151)