What are the divisors of 1951?

1, 1951

There is a total of 2 positive divisors.
The sum of these divisors is 1952.
The arithmetic mean is 976.

2 odd divisors

1, 1951

How to compute the divisors of 1951?

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 1951 by each of the numbers from 1 to 1951 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)

1951 / 1 = 1951 (the remainder is 0, so 1 is a divisor of 1951)
1951 / 2 = 975.5 (the remainder is 1, so 2 is not a divisor of 1951)
1951 / 3 = 650.33333333333 (the remainder is 1, so 3 is not a divisor of 1951)
...
1951 / 1950 = 1.0005128205128 (the remainder is 1, so 1950 is not a divisor of 1951)
1951 / 1951 = 1 (the remainder is 0, so 1951 is a divisor of 1951)

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 1951 (i.e. 44.170125650716). 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 $\frac{N}{d} = D$ )

1951 / 1 = 1951 (the remainder is 0, so 1 and 1951 are divisors of 1951)
1951 / 2 = 975.5 (the remainder is 1, so 2 is not a divisor of 1951)
1951 / 3 = 650.33333333333 (the remainder is 1, so 3 is not a divisor of 1951)
...
1951 / 43 = 45.372093023256 (the remainder is 16, so 43 is not a divisor of 1951)
1951 / 44 = 44.340909090909 (the remainder is 15, so 44 is not a divisor of 1951)