What are the divisors of 1962?

1, 2, 3, 6, 9, 18, 109, 218, 327, 654, 981, 1962

There is a total of 12 positive divisors.
The sum of these divisors is 4290.
The arithmetic mean is 357.5.

6 even divisors

2, 6, 18, 218, 654, 1962

6 odd divisors

1, 3, 9, 109, 327, 981

How to compute the divisors of 1962?

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 1962 by each of the numbers from 1 to 1962 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)

1962 / 1 = 1962 (the remainder is 0, so 1 is a divisor of 1962)
1962 / 2 = 981 (the remainder is 0, so 2 is a divisor of 1962)
1962 / 3 = 654 (the remainder is 0, so 3 is a divisor of 1962)
...
1962 / 1961 = 1.0005099439062 (the remainder is 1, so 1961 is not a divisor of 1962)
1962 / 1962 = 1 (the remainder is 0, so 1962 is a divisor of 1962)

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 1962 (i.e. 44.2944691807). 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 $\frac{N}{d} = D$ )

1962 / 1 = 1962 (the remainder is 0, so 1 and 1962 are divisors of 1962)
1962 / 2 = 981 (the remainder is 0, so 2 and 981 are divisors of 1962)
1962 / 3 = 654 (the remainder is 0, so 3 and 654 are divisors of 1962)
...
1962 / 43 = 45.627906976744 (the remainder is 27, so 43 is not a divisor of 1962)
1962 / 44 = 44.590909090909 (the remainder is 26, so 44 is not a divisor of 1962)