What are the divisors of 1401?

1, 3, 467, 1401

There is a total of 4 positive divisors.
The sum of these divisors is 1872.
The arithmetic mean is 468.

4 odd divisors

1, 3, 467, 1401

How to compute the divisors of 1401?

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 1401 by each of the numbers from 1 to 1401 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)

1401 / 1 = 1401 (the remainder is 0, so 1 is a divisor of 1401)
1401 / 2 = 700.5 (the remainder is 1, so 2 is not a divisor of 1401)
1401 / 3 = 467 (the remainder is 0, so 3 is a divisor of 1401)
...
1401 / 1400 = 1.0007142857143 (the remainder is 1, so 1400 is not a divisor of 1401)
1401 / 1401 = 1 (the remainder is 0, so 1401 is a divisor of 1401)

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 1401 (i.e. 37.429934544426). 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 $\frac{N}{d} = D$ )

1401 / 1 = 1401 (the remainder is 0, so 1 and 1401 are divisors of 1401)
1401 / 2 = 700.5 (the remainder is 1, so 2 is not a divisor of 1401)
1401 / 3 = 467 (the remainder is 0, so 3 and 467 are divisors of 1401)
...
1401 / 36 = 38.916666666667 (the remainder is 33, so 36 is not a divisor of 1401)
1401 / 37 = 37.864864864865 (the remainder is 32, so 37 is not a divisor of 1401)