What are the divisors of 4974?

1, 2, 3, 6, 829, 1658, 2487, 4974

There is a total of 8 positive divisors.
The sum of these divisors is 9960.
The arithmetic mean is 1245.

4 even divisors

2, 6, 1658, 4974

4 odd divisors

1, 3, 829, 2487

How to compute the divisors of 4974?

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 4974 by each of the numbers from 1 to 4974 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)

4974 / 1 = 4974 (the remainder is 0, so 1 is a divisor of 4974)
4974 / 2 = 2487 (the remainder is 0, so 2 is a divisor of 4974)
4974 / 3 = 1658 (the remainder is 0, so 3 is a divisor of 4974)
...
4974 / 4973 = 1.0002010858637 (the remainder is 1, so 4973 is not a divisor of 4974)
4974 / 4974 = 1 (the remainder is 0, so 4974 is a divisor of 4974)

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 4974 (i.e. 70.526590730022). 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 $\frac{N}{d} = D$ )

4974 / 1 = 4974 (the remainder is 0, so 1 and 4974 are divisors of 4974)
4974 / 2 = 2487 (the remainder is 0, so 2 and 2487 are divisors of 4974)
4974 / 3 = 1658 (the remainder is 0, so 3 and 1658 are divisors of 4974)
...
4974 / 69 = 72.086956521739 (the remainder is 6, so 69 is not a divisor of 4974)
4974 / 70 = 71.057142857143 (the remainder is 4, so 70 is not a divisor of 4974)