What are the divisors of 153?

1, 3, 9, 17, 51, 153

There is a total of 6 positive divisors.
The sum of these divisors is 234.
The arithmetic mean is 39.

6 odd divisors

1, 3, 9, 17, 51, 153

How to compute the divisors of 153?

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 153 by each of the numbers from 1 to 153 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)

153 / 1 = 153 (the remainder is 0, so 1 is a divisor of 153)
153 / 2 = 76.5 (the remainder is 1, so 2 is not a divisor of 153)
153 / 3 = 51 (the remainder is 0, so 3 is a divisor of 153)
...
153 / 152 = 1.0065789473684 (the remainder is 1, so 152 is not a divisor of 153)
153 / 153 = 1 (the remainder is 0, so 153 is a divisor of 153)

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 153 (i.e. 12.369316876853). 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 $\frac{N}{d} = D$ )

153 / 1 = 153 (the remainder is 0, so 1 and 153 are divisors of 153)
153 / 2 = 76.5 (the remainder is 1, so 2 is not a divisor of 153)
153 / 3 = 51 (the remainder is 0, so 3 and 51 are divisors of 153)
...
153 / 11 = 13.909090909091 (the remainder is 10, so 11 is not a divisor of 153)
153 / 12 = 12.75 (the remainder is 9, so 12 is not a divisor of 153)