What are the divisors of 1097?

1, 1097

There is a total of 2 positive divisors.
The sum of these divisors is 1098.
The arithmetic mean is 549.

2 odd divisors

1, 1097

How to compute the divisors of 1097?

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 1097 by each of the numbers from 1 to 1097 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)

1097 / 1 = 1097 (the remainder is 0, so 1 is a divisor of 1097)
1097 / 2 = 548.5 (the remainder is 1, so 2 is not a divisor of 1097)
1097 / 3 = 365.66666666667 (the remainder is 2, so 3 is not a divisor of 1097)
...
1097 / 1096 = 1.0009124087591 (the remainder is 1, so 1096 is not a divisor of 1097)
1097 / 1097 = 1 (the remainder is 0, so 1097 is a divisor of 1097)

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 1097 (i.e. 33.120990323358). 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 $\frac{N}{d} = D$ )

1097 / 1 = 1097 (the remainder is 0, so 1 and 1097 are divisors of 1097)
1097 / 2 = 548.5 (the remainder is 1, so 2 is not a divisor of 1097)
1097 / 3 = 365.66666666667 (the remainder is 2, so 3 is not a divisor of 1097)
...
1097 / 32 = 34.28125 (the remainder is 9, so 32 is not a divisor of 1097)
1097 / 33 = 33.242424242424 (the remainder is 8, so 33 is not a divisor of 1097)