What are the divisors of 1451?

1, 1451

There is a total of 2 positive divisors.
The sum of these divisors is 1452.
The arithmetic mean is 726.

2 odd divisors

1, 1451

How to compute the divisors of 1451?

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 1451 by each of the numbers from 1 to 1451 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)

1451 / 1 = 1451 (the remainder is 0, so 1 is a divisor of 1451)
1451 / 2 = 725.5 (the remainder is 1, so 2 is not a divisor of 1451)
1451 / 3 = 483.66666666667 (the remainder is 2, so 3 is not a divisor of 1451)
...
1451 / 1450 = 1.0006896551724 (the remainder is 1, so 1450 is not a divisor of 1451)
1451 / 1451 = 1 (the remainder is 0, so 1451 is a divisor of 1451)

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 1451 (i.e. 38.091993909482). 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 $\frac{N}{d} = D$ )

1451 / 1 = 1451 (the remainder is 0, so 1 and 1451 are divisors of 1451)
1451 / 2 = 725.5 (the remainder is 1, so 2 is not a divisor of 1451)
1451 / 3 = 483.66666666667 (the remainder is 2, so 3 is not a divisor of 1451)
...
1451 / 37 = 39.216216216216 (the remainder is 8, so 37 is not a divisor of 1451)
1451 / 38 = 38.184210526316 (the remainder is 7, so 38 is not a divisor of 1451)