What are the divisors of 4971?

1, 3, 1657, 4971

There is a total of 4 positive divisors.
The sum of these divisors is 6632.
The arithmetic mean is 1658.

4 odd divisors

1, 3, 1657, 4971

How to compute the divisors of 4971?

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 4971 by each of the numbers from 1 to 4971 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)

4971 / 1 = 4971 (the remainder is 0, so 1 is a divisor of 4971)
4971 / 2 = 2485.5 (the remainder is 1, so 2 is not a divisor of 4971)
4971 / 3 = 1657 (the remainder is 0, so 3 is a divisor of 4971)
...
4971 / 4970 = 1.0002012072435 (the remainder is 1, so 4970 is not a divisor of 4971)
4971 / 4971 = 1 (the remainder is 0, so 4971 is a divisor of 4971)

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 4971 (i.e. 70.505318948289). 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 $\frac{N}{d} = D$ )

4971 / 1 = 4971 (the remainder is 0, so 1 and 4971 are divisors of 4971)
4971 / 2 = 2485.5 (the remainder is 1, so 2 is not a divisor of 4971)
4971 / 3 = 1657 (the remainder is 0, so 3 and 1657 are divisors of 4971)
...
4971 / 69 = 72.04347826087 (the remainder is 3, so 69 is not a divisor of 4971)
4971 / 70 = 71.014285714286 (the remainder is 1, so 70 is not a divisor of 4971)