What are the divisors of 1223?

1, 1223

There is a total of 2 positive divisors.
The sum of these divisors is 1224.
The arithmetic mean is 612.

2 odd divisors

1, 1223

How to compute the divisors of 1223?

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 1223 by each of the numbers from 1 to 1223 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)

1223 / 1 = 1223 (the remainder is 0, so 1 is a divisor of 1223)
1223 / 2 = 611.5 (the remainder is 1, so 2 is not a divisor of 1223)
1223 / 3 = 407.66666666667 (the remainder is 2, so 3 is not a divisor of 1223)
...
1223 / 1222 = 1.0008183306056 (the remainder is 1, so 1222 is not a divisor of 1223)
1223 / 1223 = 1 (the remainder is 0, so 1223 is a divisor of 1223)

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 1223 (i.e. 34.971416900091). 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 $\frac{N}{d} = D$ )

1223 / 1 = 1223 (the remainder is 0, so 1 and 1223 are divisors of 1223)
1223 / 2 = 611.5 (the remainder is 1, so 2 is not a divisor of 1223)
1223 / 3 = 407.66666666667 (the remainder is 2, so 3 is not a divisor of 1223)
...
1223 / 33 = 37.060606060606 (the remainder is 2, so 33 is not a divisor of 1223)
1223 / 34 = 35.970588235294 (the remainder is 33, so 34 is not a divisor of 1223)