What are the divisors of 1947?

1, 3, 11, 33, 59, 177, 649, 1947

There is a total of 8 positive divisors.
The sum of these divisors is 2880.
The arithmetic mean is 360.

8 odd divisors

1, 3, 11, 33, 59, 177, 649, 1947

How to compute the divisors of 1947?

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 1947 by each of the numbers from 1 to 1947 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)

1947 / 1 = 1947 (the remainder is 0, so 1 is a divisor of 1947)
1947 / 2 = 973.5 (the remainder is 1, so 2 is not a divisor of 1947)
1947 / 3 = 649 (the remainder is 0, so 3 is a divisor of 1947)
...
1947 / 1946 = 1.0005138746146 (the remainder is 1, so 1946 is not a divisor of 1947)
1947 / 1947 = 1 (the remainder is 0, so 1947 is a divisor of 1947)

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 1947 (i.e. 44.12482294582). 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 $\frac{N}{d} = D$ )

1947 / 1 = 1947 (the remainder is 0, so 1 and 1947 are divisors of 1947)
1947 / 2 = 973.5 (the remainder is 1, so 2 is not a divisor of 1947)
1947 / 3 = 649 (the remainder is 0, so 3 and 649 are divisors of 1947)
...
1947 / 43 = 45.279069767442 (the remainder is 12, so 43 is not a divisor of 1947)
1947 / 44 = 44.25 (the remainder is 11, so 44 is not a divisor of 1947)