What are the divisors of 145?

1, 5, 29, 145

There is a total of 4 positive divisors.
The sum of these divisors is 180.
The arithmetic mean is 45.

4 odd divisors

1, 5, 29, 145

How to compute the divisors of 145?

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 145 by each of the numbers from 1 to 145 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)

145 / 1 = 145 (the remainder is 0, so 1 is a divisor of 145)
145 / 2 = 72.5 (the remainder is 1, so 2 is not a divisor of 145)
145 / 3 = 48.333333333333 (the remainder is 1, so 3 is not a divisor of 145)
...
145 / 144 = 1.0069444444444 (the remainder is 1, so 144 is not a divisor of 145)
145 / 145 = 1 (the remainder is 0, so 145 is a divisor of 145)

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 145 (i.e. 12.041594578792). 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 $\frac{N}{d} = D$ )

145 / 1 = 145 (the remainder is 0, so 1 and 145 are divisors of 145)
145 / 2 = 72.5 (the remainder is 1, so 2 is not a divisor of 145)
145 / 3 = 48.333333333333 (the remainder is 1, so 3 is not a divisor of 145)
...
145 / 11 = 13.181818181818 (the remainder is 2, so 11 is not a divisor of 145)
145 / 12 = 12.083333333333 (the remainder is 1, so 12 is not a divisor of 145)