What are the divisors of 137?

1, 137

There is a total of 2 positive divisors.
The sum of these divisors is 138.
The arithmetic mean is 69.

2 odd divisors

1, 137

How to compute the divisors of 137?

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 137 by each of the numbers from 1 to 137 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)

137 / 1 = 137 (the remainder is 0, so 1 is a divisor of 137)
137 / 2 = 68.5 (the remainder is 1, so 2 is not a divisor of 137)
137 / 3 = 45.666666666667 (the remainder is 2, so 3 is not a divisor of 137)
...
137 / 136 = 1.0073529411765 (the remainder is 1, so 136 is not a divisor of 137)
137 / 137 = 1 (the remainder is 0, so 137 is a divisor of 137)

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 137 (i.e. 11.70469991072). 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 $\frac{N}{d} = D$ )

137 / 1 = 137 (the remainder is 0, so 1 and 137 are divisors of 137)
137 / 2 = 68.5 (the remainder is 1, so 2 is not a divisor of 137)
137 / 3 = 45.666666666667 (the remainder is 2, so 3 is not a divisor of 137)
...
137 / 10 = 13.7 (the remainder is 7, so 10 is not a divisor of 137)
137 / 11 = 12.454545454545 (the remainder is 5, so 11 is not a divisor of 137)