What are the divisors of 193?

1, 193

There is a total of 2 positive divisors.
The sum of these divisors is 194.
The arithmetic mean is 97.

2 odd divisors

1, 193

How to compute the divisors of 193?

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 193 by each of the numbers from 1 to 193 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)

193 / 1 = 193 (the remainder is 0, so 1 is a divisor of 193)
193 / 2 = 96.5 (the remainder is 1, so 2 is not a divisor of 193)
193 / 3 = 64.333333333333 (the remainder is 1, so 3 is not a divisor of 193)
...
193 / 192 = 1.0052083333333 (the remainder is 1, so 192 is not a divisor of 193)
193 / 193 = 1 (the remainder is 0, so 193 is a divisor of 193)

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 193 (i.e. 13.89244398945). 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 $\frac{N}{d} = D$ )

193 / 1 = 193 (the remainder is 0, so 1 and 193 are divisors of 193)
193 / 2 = 96.5 (the remainder is 1, so 2 is not a divisor of 193)
193 / 3 = 64.333333333333 (the remainder is 1, so 3 is not a divisor of 193)
...
193 / 12 = 16.083333333333 (the remainder is 1, so 12 is not a divisor of 193)
193 / 13 = 14.846153846154 (the remainder is 11, so 13 is not a divisor of 193)