What are the divisors of 1261?

1, 13, 97, 1261

There is a total of 4 positive divisors.
The sum of these divisors is 1372.
The arithmetic mean is 343.

4 odd divisors

1, 13, 97, 1261

How to compute the divisors of 1261?

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 1261 by each of the numbers from 1 to 1261 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)

1261 / 1 = 1261 (the remainder is 0, so 1 is a divisor of 1261)
1261 / 2 = 630.5 (the remainder is 1, so 2 is not a divisor of 1261)
1261 / 3 = 420.33333333333 (the remainder is 1, so 3 is not a divisor of 1261)
...
1261 / 1260 = 1.0007936507937 (the remainder is 1, so 1260 is not a divisor of 1261)
1261 / 1261 = 1 (the remainder is 0, so 1261 is a divisor of 1261)

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 1261 (i.e. 35.510561809129). 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 $\frac{N}{d} = D$ )

1261 / 1 = 1261 (the remainder is 0, so 1 and 1261 are divisors of 1261)
1261 / 2 = 630.5 (the remainder is 1, so 2 is not a divisor of 1261)
1261 / 3 = 420.33333333333 (the remainder is 1, so 3 is not a divisor of 1261)
...
1261 / 34 = 37.088235294118 (the remainder is 3, so 34 is not a divisor of 1261)
1261 / 35 = 36.028571428571 (the remainder is 1, so 35 is not a divisor of 1261)