What are the divisors of 1961?

1, 37, 53, 1961

There is a total of 4 positive divisors.
The sum of these divisors is 2052.
The arithmetic mean is 513.

4 odd divisors

1, 37, 53, 1961

How to compute the divisors of 1961?

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 1961 by each of the numbers from 1 to 1961 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)

1961 / 1 = 1961 (the remainder is 0, so 1 is a divisor of 1961)
1961 / 2 = 980.5 (the remainder is 1, so 2 is not a divisor of 1961)
1961 / 3 = 653.66666666667 (the remainder is 2, so 3 is not a divisor of 1961)
...
1961 / 1960 = 1.0005102040816 (the remainder is 1, so 1960 is not a divisor of 1961)
1961 / 1961 = 1 (the remainder is 0, so 1961 is a divisor of 1961)

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 1961 (i.e. 44.283179650969). 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 $\frac{N}{d} = D$ )

1961 / 1 = 1961 (the remainder is 0, so 1 and 1961 are divisors of 1961)
1961 / 2 = 980.5 (the remainder is 1, so 2 is not a divisor of 1961)
1961 / 3 = 653.66666666667 (the remainder is 2, so 3 is not a divisor of 1961)
...
1961 / 43 = 45.604651162791 (the remainder is 26, so 43 is not a divisor of 1961)
1961 / 44 = 44.568181818182 (the remainder is 25, so 44 is not a divisor of 1961)