What are the divisors of 1277?

1, 1277

There is a total of 2 positive divisors.
The sum of these divisors is 1278.
The arithmetic mean is 639.

2 odd divisors

1, 1277

How to compute the divisors of 1277?

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 1277 by each of the numbers from 1 to 1277 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)

1277 / 1 = 1277 (the remainder is 0, so 1 is a divisor of 1277)
1277 / 2 = 638.5 (the remainder is 1, so 2 is not a divisor of 1277)
1277 / 3 = 425.66666666667 (the remainder is 2, so 3 is not a divisor of 1277)
...
1277 / 1276 = 1.0007836990596 (the remainder is 1, so 1276 is not a divisor of 1277)
1277 / 1277 = 1 (the remainder is 0, so 1277 is a divisor of 1277)

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 1277 (i.e. 35.735136770411). 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 $\frac{N}{d} = D$ )

1277 / 1 = 1277 (the remainder is 0, so 1 and 1277 are divisors of 1277)
1277 / 2 = 638.5 (the remainder is 1, so 2 is not a divisor of 1277)
1277 / 3 = 425.66666666667 (the remainder is 2, so 3 is not a divisor of 1277)
...
1277 / 34 = 37.558823529412 (the remainder is 19, so 34 is not a divisor of 1277)
1277 / 35 = 36.485714285714 (the remainder is 17, so 35 is not a divisor of 1277)