What are the divisors of 144?

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144

There is a total of 15 positive divisors.
The sum of these divisors is 403.
The arithmetic mean is 26.866666666667.

12 even divisors

2, 4, 6, 8, 12, 16, 18, 24, 36, 48, 72, 144

3 odd divisors

1, 3, 9

How to compute the divisors of 144?

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 144 by each of the numbers from 1 to 144 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)

144 / 1 = 144 (the remainder is 0, so 1 is a divisor of 144)
144 / 2 = 72 (the remainder is 0, so 2 is a divisor of 144)
144 / 3 = 48 (the remainder is 0, so 3 is a divisor of 144)
...
144 / 143 = 1.006993006993 (the remainder is 1, so 143 is not a divisor of 144)
144 / 144 = 1 (the remainder is 0, so 144 is a divisor of 144)

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 144 (i.e. 12). 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 $\frac{N}{d} = D$ )

144 / 1 = 144 (the remainder is 0, so 1 and 144 are divisors of 144)
144 / 2 = 72 (the remainder is 0, so 2 and 72 are divisors of 144)
144 / 3 = 48 (the remainder is 0, so 3 and 48 are divisors of 144)
...
144 / 11 = 13.090909090909 (the remainder is 1, so 11 is not a divisor of 144)
144 / 12 = 12 (the remainder is 0, so 12 and 12 are divisors of 144)