What are the divisors of 4151?

1, 7, 593, 4151

There is a total of 4 positive divisors.
The sum of these divisors is 4752.
The arithmetic mean is 1188.

4 odd divisors

1, 7, 593, 4151

How to compute the divisors of 4151?

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 4151 by each of the numbers from 1 to 4151 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)

4151 / 1 = 4151 (the remainder is 0, so 1 is a divisor of 4151)
4151 / 2 = 2075.5 (the remainder is 1, so 2 is not a divisor of 4151)
4151 / 3 = 1383.6666666667 (the remainder is 2, so 3 is not a divisor of 4151)
...
4151 / 4150 = 1.0002409638554 (the remainder is 1, so 4150 is not a divisor of 4151)
4151 / 4151 = 1 (the remainder is 0, so 4151 is a divisor of 4151)

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 4151 (i.e. 64.428254671378). 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 $\frac{N}{d} = D$ )

4151 / 1 = 4151 (the remainder is 0, so 1 and 4151 are divisors of 4151)
4151 / 2 = 2075.5 (the remainder is 1, so 2 is not a divisor of 4151)
4151 / 3 = 1383.6666666667 (the remainder is 2, so 3 is not a divisor of 4151)
...
4151 / 63 = 65.888888888889 (the remainder is 56, so 63 is not a divisor of 4151)
4151 / 64 = 64.859375 (the remainder is 55, so 64 is not a divisor of 4151)