What are the divisors of 147?

1, 3, 7, 21, 49, 147

There is a total of 6 positive divisors.
The sum of these divisors is 228.
The arithmetic mean is 38.

6 odd divisors

1, 3, 7, 21, 49, 147

How to compute the divisors of 147?

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 147 by each of the numbers from 1 to 147 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)

147 / 1 = 147 (the remainder is 0, so 1 is a divisor of 147)
147 / 2 = 73.5 (the remainder is 1, so 2 is not a divisor of 147)
147 / 3 = 49 (the remainder is 0, so 3 is a divisor of 147)
...
147 / 146 = 1.0068493150685 (the remainder is 1, so 146 is not a divisor of 147)
147 / 147 = 1 (the remainder is 0, so 147 is a divisor of 147)

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 147 (i.e. 12.124355652982). 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 $\frac{N}{d} = D$ )

147 / 1 = 147 (the remainder is 0, so 1 and 147 are divisors of 147)
147 / 2 = 73.5 (the remainder is 1, so 2 is not a divisor of 147)
147 / 3 = 49 (the remainder is 0, so 3 and 49 are divisors of 147)
...
147 / 11 = 13.363636363636 (the remainder is 4, so 11 is not a divisor of 147)
147 / 12 = 12.25 (the remainder is 3, so 12 is not a divisor of 147)