What are the divisors of 1964?

1, 2, 4, 491, 982, 1964

There is a total of 6 positive divisors.
The sum of these divisors is 3444.
The arithmetic mean is 574.

4 even divisors

2, 4, 982, 1964

2 odd divisors

1, 491

How to compute the divisors of 1964?

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 1964 by each of the numbers from 1 to 1964 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)

1964 / 1 = 1964 (the remainder is 0, so 1 is a divisor of 1964)
1964 / 2 = 982 (the remainder is 0, so 2 is a divisor of 1964)
1964 / 3 = 654.66666666667 (the remainder is 2, so 3 is not a divisor of 1964)
...
1964 / 1963 = 1.0005094243505 (the remainder is 1, so 1963 is not a divisor of 1964)
1964 / 1964 = 1 (the remainder is 0, so 1964 is a divisor of 1964)

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 1964 (i.e. 44.317039612321). 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 $\frac{N}{d} = D$ )

1964 / 1 = 1964 (the remainder is 0, so 1 and 1964 are divisors of 1964)
1964 / 2 = 982 (the remainder is 0, so 2 and 982 are divisors of 1964)
1964 / 3 = 654.66666666667 (the remainder is 2, so 3 is not a divisor of 1964)
...
1964 / 43 = 45.674418604651 (the remainder is 29, so 43 is not a divisor of 1964)
1964 / 44 = 44.636363636364 (the remainder is 28, so 44 is not a divisor of 1964)