What are the divisors of 1937?

1, 13, 149, 1937

There is a total of 4 positive divisors.
The sum of these divisors is 2100.
The arithmetic mean is 525.

4 odd divisors

1, 13, 149, 1937

How to compute the divisors of 1937?

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 1937 by each of the numbers from 1 to 1937 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)

1937 / 1 = 1937 (the remainder is 0, so 1 is a divisor of 1937)
1937 / 2 = 968.5 (the remainder is 1, so 2 is not a divisor of 1937)
1937 / 3 = 645.66666666667 (the remainder is 2, so 3 is not a divisor of 1937)
...
1937 / 1936 = 1.0005165289256 (the remainder is 1, so 1936 is not a divisor of 1937)
1937 / 1937 = 1 (the remainder is 0, so 1937 is a divisor of 1937)

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 1937 (i.e. 44.011362169331). 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 $\frac{N}{d} = D$ )

1937 / 1 = 1937 (the remainder is 0, so 1 and 1937 are divisors of 1937)
1937 / 2 = 968.5 (the remainder is 1, so 2 is not a divisor of 1937)
1937 / 3 = 645.66666666667 (the remainder is 2, so 3 is not a divisor of 1937)
...
1937 / 43 = 45.046511627907 (the remainder is 2, so 43 is not a divisor of 1937)
1937 / 44 = 44.022727272727 (the remainder is 1, so 44 is not a divisor of 1937)