What are the divisors of 1213?

1, 1213

There is a total of 2 positive divisors.
The sum of these divisors is 1214.
The arithmetic mean is 607.

2 odd divisors

1, 1213

How to compute the divisors of 1213?

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 1213 by each of the numbers from 1 to 1213 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)

1213 / 1 = 1213 (the remainder is 0, so 1 is a divisor of 1213)
1213 / 2 = 606.5 (the remainder is 1, so 2 is not a divisor of 1213)
1213 / 3 = 404.33333333333 (the remainder is 1, so 3 is not a divisor of 1213)
...
1213 / 1212 = 1.0008250825083 (the remainder is 1, so 1212 is not a divisor of 1213)
1213 / 1213 = 1 (the remainder is 0, so 1213 is a divisor of 1213)

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 1213 (i.e. 34.828149534536). 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 $\frac{N}{d} = D$ )

1213 / 1 = 1213 (the remainder is 0, so 1 and 1213 are divisors of 1213)
1213 / 2 = 606.5 (the remainder is 1, so 2 is not a divisor of 1213)
1213 / 3 = 404.33333333333 (the remainder is 1, so 3 is not a divisor of 1213)
...
1213 / 33 = 36.757575757576 (the remainder is 25, so 33 is not a divisor of 1213)
1213 / 34 = 35.676470588235 (the remainder is 23, so 34 is not a divisor of 1213)