What are the divisors of 169?

1, 13, 169

There is a total of 3 positive divisors.
The sum of these divisors is 183.
The arithmetic mean is 61.

3 odd divisors

1, 13, 169

How to compute the divisors of 169?

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 169 by each of the numbers from 1 to 169 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)

169 / 1 = 169 (the remainder is 0, so 1 is a divisor of 169)
169 / 2 = 84.5 (the remainder is 1, so 2 is not a divisor of 169)
169 / 3 = 56.333333333333 (the remainder is 1, so 3 is not a divisor of 169)
...
169 / 168 = 1.0059523809524 (the remainder is 1, so 168 is not a divisor of 169)
169 / 169 = 1 (the remainder is 0, so 169 is a divisor of 169)

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 169 (i.e. 13). 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 $\frac{N}{d} = D$ )

169 / 1 = 169 (the remainder is 0, so 1 and 169 are divisors of 169)
169 / 2 = 84.5 (the remainder is 1, so 2 is not a divisor of 169)
169 / 3 = 56.333333333333 (the remainder is 1, so 3 is not a divisor of 169)
...
169 / 12 = 14.083333333333 (the remainder is 1, so 12 is not a divisor of 169)
169 / 13 = 13 (the remainder is 0, so 13 and 13 are divisors of 169)