What are the divisors of 1050?

1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42, 50, 70, 75, 105, 150, 175, 210, 350, 525, 1050

There is a total of 24 positive divisors.
The sum of these divisors is 2976.
The arithmetic mean is 124.

12 even divisors

2, 6, 10, 14, 30, 42, 50, 70, 150, 210, 350, 1050

12 odd divisors

1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525

How to compute the divisors of 1050?

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 1050 by each of the numbers from 1 to 1050 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)

1050 / 1 = 1050 (the remainder is 0, so 1 is a divisor of 1050)
1050 / 2 = 525 (the remainder is 0, so 2 is a divisor of 1050)
1050 / 3 = 350 (the remainder is 0, so 3 is a divisor of 1050)
...
1050 / 1049 = 1.0009532888465 (the remainder is 1, so 1049 is not a divisor of 1050)
1050 / 1050 = 1 (the remainder is 0, so 1050 is a divisor of 1050)

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 1050 (i.e. 32.403703492039). 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 $\frac{N}{d} = D$ )

1050 / 1 = 1050 (the remainder is 0, so 1 and 1050 are divisors of 1050)
1050 / 2 = 525 (the remainder is 0, so 2 and 525 are divisors of 1050)
1050 / 3 = 350 (the remainder is 0, so 3 and 350 are divisors of 1050)
...
1050 / 31 = 33.870967741935 (the remainder is 27, so 31 is not a divisor of 1050)
1050 / 32 = 32.8125 (the remainder is 26, so 32 is not a divisor of 1050)