What are the divisors of 1949?

1, 1949

There is a total of 2 positive divisors.
The sum of these divisors is 1950.
The arithmetic mean is 975.

2 odd divisors

1, 1949

How to compute the divisors of 1949?

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 1949 by each of the numbers from 1 to 1949 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)

1949 / 1 = 1949 (the remainder is 0, so 1 is a divisor of 1949)
1949 / 2 = 974.5 (the remainder is 1, so 2 is not a divisor of 1949)
1949 / 3 = 649.66666666667 (the remainder is 2, so 3 is not a divisor of 1949)
...
1949 / 1948 = 1.0005133470226 (the remainder is 1, so 1948 is not a divisor of 1949)
1949 / 1949 = 1 (the remainder is 0, so 1949 is a divisor of 1949)

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 1949 (i.e. 44.147480109288). 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 $\frac{N}{d} = D$ )

1949 / 1 = 1949 (the remainder is 0, so 1 and 1949 are divisors of 1949)
1949 / 2 = 974.5 (the remainder is 1, so 2 is not a divisor of 1949)
1949 / 3 = 649.66666666667 (the remainder is 2, so 3 is not a divisor of 1949)
...
1949 / 43 = 45.325581395349 (the remainder is 14, so 43 is not a divisor of 1949)
1949 / 44 = 44.295454545455 (the remainder is 13, so 44 is not a divisor of 1949)