What are the divisors of 1257?

1, 3, 419, 1257

There is a total of 4 positive divisors.
The sum of these divisors is 1680.
The arithmetic mean is 420.

4 odd divisors

1, 3, 419, 1257

How to compute the divisors of 1257?

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 1257 by each of the numbers from 1 to 1257 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)

1257 / 1 = 1257 (the remainder is 0, so 1 is a divisor of 1257)
1257 / 2 = 628.5 (the remainder is 1, so 2 is not a divisor of 1257)
1257 / 3 = 419 (the remainder is 0, so 3 is a divisor of 1257)
...
1257 / 1256 = 1.0007961783439 (the remainder is 1, so 1256 is not a divisor of 1257)
1257 / 1257 = 1 (the remainder is 0, so 1257 is a divisor of 1257)

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 1257 (i.e. 35.454195802472). 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 $\frac{N}{d} = D$ )

1257 / 1 = 1257 (the remainder is 0, so 1 and 1257 are divisors of 1257)
1257 / 2 = 628.5 (the remainder is 1, so 2 is not a divisor of 1257)
1257 / 3 = 419 (the remainder is 0, so 3 and 419 are divisors of 1257)
...
1257 / 34 = 36.970588235294 (the remainder is 33, so 34 is not a divisor of 1257)
1257 / 35 = 35.914285714286 (the remainder is 32, so 35 is not a divisor of 1257)