What are the divisors of 401?

1, 401

There is a total of 2 positive divisors.
The sum of these divisors is 402.
The arithmetic mean is 201.

2 odd divisors

1, 401

How to compute the divisors of 401?

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 401 by each of the numbers from 1 to 401 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)

401 / 1 = 401 (the remainder is 0, so 1 is a divisor of 401)
401 / 2 = 200.5 (the remainder is 1, so 2 is not a divisor of 401)
401 / 3 = 133.66666666667 (the remainder is 2, so 3 is not a divisor of 401)
...
401 / 400 = 1.0025 (the remainder is 1, so 400 is not a divisor of 401)
401 / 401 = 1 (the remainder is 0, so 401 is a divisor of 401)

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 401 (i.e. 20.024984394501). 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 $\frac{N}{d} = D$ )

401 / 1 = 401 (the remainder is 0, so 1 and 401 are divisors of 401)
401 / 2 = 200.5 (the remainder is 1, so 2 is not a divisor of 401)
401 / 3 = 133.66666666667 (the remainder is 2, so 3 is not a divisor of 401)
...
401 / 19 = 21.105263157895 (the remainder is 2, so 19 is not a divisor of 401)
401 / 20 = 20.05 (the remainder is 1, so 20 is not a divisor of 401)