What are the divisors of 1273?

1, 19, 67, 1273

There is a total of 4 positive divisors.
The sum of these divisors is 1360.
The arithmetic mean is 340.

4 odd divisors

1, 19, 67, 1273

How to compute the divisors of 1273?

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 1273 by each of the numbers from 1 to 1273 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)

1273 / 1 = 1273 (the remainder is 0, so 1 is a divisor of 1273)
1273 / 2 = 636.5 (the remainder is 1, so 2 is not a divisor of 1273)
1273 / 3 = 424.33333333333 (the remainder is 1, so 3 is not a divisor of 1273)
...
1273 / 1272 = 1.000786163522 (the remainder is 1, so 1272 is not a divisor of 1273)
1273 / 1273 = 1 (the remainder is 0, so 1273 is a divisor of 1273)

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 1273 (i.e. 35.679125549823). 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 $\frac{N}{d} = D$ )

1273 / 1 = 1273 (the remainder is 0, so 1 and 1273 are divisors of 1273)
1273 / 2 = 636.5 (the remainder is 1, so 2 is not a divisor of 1273)
1273 / 3 = 424.33333333333 (the remainder is 1, so 3 is not a divisor of 1273)
...
1273 / 34 = 37.441176470588 (the remainder is 15, so 34 is not a divisor of 1273)
1273 / 35 = 36.371428571429 (the remainder is 13, so 35 is not a divisor of 1273)