What are the divisors of 4972?

1, 2, 4, 11, 22, 44, 113, 226, 452, 1243, 2486, 4972

There is a total of 12 positive divisors.
The sum of these divisors is 9576.
The arithmetic mean is 798.

8 even divisors

2, 4, 22, 44, 226, 452, 2486, 4972

4 odd divisors

1, 11, 113, 1243

How to compute the divisors of 4972?

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 4972 by each of the numbers from 1 to 4972 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)

4972 / 1 = 4972 (the remainder is 0, so 1 is a divisor of 4972)
4972 / 2 = 2486 (the remainder is 0, so 2 is a divisor of 4972)
4972 / 3 = 1657.3333333333 (the remainder is 1, so 3 is not a divisor of 4972)
...
4972 / 4971 = 1.0002011667673 (the remainder is 1, so 4971 is not a divisor of 4972)
4972 / 4972 = 1 (the remainder is 0, so 4972 is a divisor of 4972)

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 4972 (i.e. 70.512410255217). 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 $\frac{N}{d} = D$ )

4972 / 1 = 4972 (the remainder is 0, so 1 and 4972 are divisors of 4972)
4972 / 2 = 2486 (the remainder is 0, so 2 and 2486 are divisors of 4972)
4972 / 3 = 1657.3333333333 (the remainder is 1, so 3 is not a divisor of 4972)
...
4972 / 69 = 72.057971014493 (the remainder is 4, so 69 is not a divisor of 4972)
4972 / 70 = 71.028571428571 (the remainder is 2, so 70 is not a divisor of 4972)