What are the divisors of 441?

1, 3, 7, 9, 21, 49, 63, 147, 441

There is a total of 9 positive divisors.
The sum of these divisors is 741.
The arithmetic mean is 82.333333333333.

9 odd divisors

1, 3, 7, 9, 21, 49, 63, 147, 441

How to compute the divisors of 441?

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 441 by each of the numbers from 1 to 441 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)

441 / 1 = 441 (the remainder is 0, so 1 is a divisor of 441)
441 / 2 = 220.5 (the remainder is 1, so 2 is not a divisor of 441)
441 / 3 = 147 (the remainder is 0, so 3 is a divisor of 441)
...
441 / 440 = 1.0022727272727 (the remainder is 1, so 440 is not a divisor of 441)
441 / 441 = 1 (the remainder is 0, so 441 is a divisor of 441)

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 441 (i.e. 21). 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 $\frac{N}{d} = D$ )

441 / 1 = 441 (the remainder is 0, so 1 and 441 are divisors of 441)
441 / 2 = 220.5 (the remainder is 1, so 2 is not a divisor of 441)
441 / 3 = 147 (the remainder is 0, so 3 and 147 are divisors of 441)
...
441 / 20 = 22.05 (the remainder is 1, so 20 is not a divisor of 441)
441 / 21 = 21 (the remainder is 0, so 21 and 21 are divisors of 441)