What are the divisors of 1337?

1, 7, 191, 1337

There is a total of 4 positive divisors.
The sum of these divisors is 1536.
The arithmetic mean is 384.

4 odd divisors

1, 7, 191, 1337

How to compute the divisors of 1337?

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 1337 by each of the numbers from 1 to 1337 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)

1337 / 1 = 1337 (the remainder is 0, so 1 is a divisor of 1337)
1337 / 2 = 668.5 (the remainder is 1, so 2 is not a divisor of 1337)
1337 / 3 = 445.66666666667 (the remainder is 2, so 3 is not a divisor of 1337)
...
1337 / 1336 = 1.000748502994 (the remainder is 1, so 1336 is not a divisor of 1337)
1337 / 1337 = 1 (the remainder is 0, so 1337 is a divisor of 1337)

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 1337 (i.e. 36.565010597564). 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 $\frac{N}{d} = D$ )

1337 / 1 = 1337 (the remainder is 0, so 1 and 1337 are divisors of 1337)
1337 / 2 = 668.5 (the remainder is 1, so 2 is not a divisor of 1337)
1337 / 3 = 445.66666666667 (the remainder is 2, so 3 is not a divisor of 1337)
...
1337 / 35 = 38.2 (the remainder is 7, so 35 is not a divisor of 1337)
1337 / 36 = 37.138888888889 (the remainder is 5, so 36 is not a divisor of 1337)