What are the divisors of 1972?

1, 2, 4, 17, 29, 34, 58, 68, 116, 493, 986, 1972

There is a total of 12 positive divisors.
The sum of these divisors is 3780.
The arithmetic mean is 315.

8 even divisors

2, 4, 34, 58, 68, 116, 986, 1972

4 odd divisors

1, 17, 29, 493

How to compute the divisors of 1972?

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 1972 by each of the numbers from 1 to 1972 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)

1972 / 1 = 1972 (the remainder is 0, so 1 is a divisor of 1972)
1972 / 2 = 986 (the remainder is 0, so 2 is a divisor of 1972)
1972 / 3 = 657.33333333333 (the remainder is 1, so 3 is not a divisor of 1972)
...
1972 / 1971 = 1.0005073566717 (the remainder is 1, so 1971 is not a divisor of 1972)
1972 / 1972 = 1 (the remainder is 0, so 1972 is a divisor of 1972)

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 1972 (i.e. 44.407206622349). 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 $\frac{N}{d} = D$ )

1972 / 1 = 1972 (the remainder is 0, so 1 and 1972 are divisors of 1972)
1972 / 2 = 986 (the remainder is 0, so 2 and 986 are divisors of 1972)
1972 / 3 = 657.33333333333 (the remainder is 1, so 3 is not a divisor of 1972)
...
1972 / 43 = 45.860465116279 (the remainder is 37, so 43 is not a divisor of 1972)
1972 / 44 = 44.818181818182 (the remainder is 36, so 44 is not a divisor of 1972)